Vishnu in His vyūha manifestation which is four-fold (Chatur-vyūha) – Vasudeva (as the creator),Samkarsana (as the sustainer),Pradyumna (as destroyer - dissolution of the universe) and Aniruddha (as promulgator of spiritual knowledge). This is the Vaiśnava doctrine of Vyūha or the doctrine of formation. The Chatur-vyūha forms of Vishnu are related to four of the six causes of creation which six are God Himself as the final cause of creation and His five aspects – Narāyana ('thinking'), Vāsudeva ('feeling'), Samkarśana ('willing'), Pradyumna ('knowing') and Aniruddha ('acting') successively; each divinity controls its specific creative energy. Vyuha
Here is the excerpt from Shanti Parva ,Mahabharata
I am Brahma, the primeval master of all creatures, have started into birth from Him, and thou hast taken thy birth from me. From me have flowed the universe with all its mobile and immobile creatures, and all the Vedas, O son, with their mysteries. Divided into four portions (viz., Aniruddha, Pradyumna, Sankarshana, and Vasudeva.)
I am giving answer about ,Samkarsana and ,Pradyumna avataras only Based on Bhagvat Purana.
Pradyumna avatara - Pradyumna was son of Lord Krishna and Rukmini Pradyumna was born in Dvaraka. He was the version of God Kamdev. In the Treta Yuga, Kamdev was burnt by Shiva when he became a barrier to Shiva's meditation. Shiva blessed Kamdev's distraught wife, Rati and promised her that in his next birth Kamdev will be a part of Krishna and Rati will be the daughter of Bhimkaraya Rukmi and that she will marry him.
कामास्तु वसुदेवांशो दग्ध: प्राग रुद्रमन्युना देहोपपत्तये भुयस्तमेव प्रत्यप स एव
जातो वैदर्भ्यां कृष्णवीर्यसमुद्भव: प्रद्युम्न इति विख्याता: सर्वतोsनवम: पितू: ||1 ,2||
kāmas tu vāsudevāṁśo dagdhaḥ prāg rudra-manyunā dehopapattaye bhūyas tam eva pratyapadyata sa eva jāto vaidarbhyāṁ kṛṣṇa-vīrya-samudbhavaḥ pradyumna iti vikhyātaḥ sarvato ’navamaḥ pituḥ
Kāmadeva [Cupid], an expansion of Vāsudeva, had previously been burned to ashes by Rudra’s anger. Now, to obtain a new body, he merged back into the body of Lord Vāsudeva. He took birth in the womb of Vaidarbhī from the seed of Lord Kṛṣṇa and received the name Pradyumna. In no respect was He inferior to His father.SB 10.55.1 and SB 10.55.2
Once, Ratidevī approached Pradyumna. She advised Him to kill Śambara, and to help Him she instructed Him in the mystic mantras known as Mahā-māyā.
तमिमं जहि दुर्धर्ष दुर्जयं शत्रुमात्मन:
मयशतविदं त्वं च मायाभिर्मोहनादिभि: ||14|
tam imaṁ jahi durdharṣaṁ durjayaṁ śatrum ātmanaḥ māyā-śata-vidaṁ taṁ ca māyābhir mohanādibhiḥ
Now kill this dreadful Śambara, Your formidable enemy. Although he knows hundreds of magic spells, You can defeat him with bewildering magic and other techniques. SB 10.55.14
Sankarshana Avatara -
तस्य मूल-देशे त्रिंशद्योजनसहस्त्रान्तर आस्ते
या वै कला भगवतस्तामसी समाख्याताsन्त इतिsसात्वतिया
दृष्टदृश्यो: संड.कर्षम् अहमित्यभिमान यं संड.कर्षम् इति आचक्षते ||1||
tasya mūla-deśe triṁśad-yojana-sahasrāntara āste yā vai kalā bhagavatas tāmasī samākhyātānanta iti sātvatīyā draṣṭṛ-dṛśyayoḥ saṅkarṣaṇam aham ity abhimāna-lakṣaṇaṁ yaṁ saṅkarṣaṇam ity ācakṣate.
Śrī Śukadeva Gosvāmī said to Mahārāja Parīkṣit: My dear King, approximately 240,000 miles beneath the planet Pātāla lives another incarnation of the Supreme Personality of Godhead. He is the expansion of Lord Viṣṇu known as Lord Ananta or Lord Saṅkarṣaṇa. He is always in the transcendental position, but because He is worshiped by Lord Śiva, the deity of tamo-guṇa or darkness, He is sometimes called tāmasī. Lord Ananta is the predominating Deity of the material mode of ignorance as well as the false ego of all conditioned souls. When a conditioned living being thinks, “I am the enjoyer, and this world is meant to be enjoyed by me,” this conception of life is dictated to him by Saṅkarṣaṇa. Thus the mundane conditioned soul thinks himself the Supreme Lord.SB 5.25.1
यस्य ह वा इदं कालेनोपसंजिहीर्षतोsमर्ष – विरचितरुचिरभ्रमद्भ्रुवोरन्तरेण
साडक.र्षणो नाम रुद्र एकादशव्यूहस्त्र्यक्षस्त्रिशिखं शूलमुत्तम्भ्ययन्नु-दतिषठत् ||3||
yasya ha vā idaṁ kālenopasañjihīrṣato ’marṣa-viracita-rucira-bhramad-bhruvor antareṇa sāṅkarṣaṇo nāma rudra ekādaśa-vyūhas try-akṣas tri-śikhaṁ śūlam uttambhayann udatiṣṭhat.
At the time of devastation, when Lord Anantadeva desires to destroy the entire creation, He becomes slightly angry. Then from between His two eyebrows appears three-eyed Rudra, carrying a trident. This Rudra, who is known as Sāṅkarṣaṇa, is the embodiment of the eleven Rudras, or incarnations of Lord Śiva. He appears in order to devastate the entire creation.SB 5.25.3
So from these shlokas we come to know that Sankarshna avatar takes place , when lord wishes to destroy the word ,as Sankarshna Rudra.
Your answers are all given in Srimad Bhagavatam.
My Lord, as the supreme directing Deity known as Aniruddha, You are the master of the senses and the mind. I therefore offer my obeisances unto You again and again. You are known as Ananta as well as Saṅkarṣaṇa because of Your ability to destroy the whole creation by the blazing fire from Your mouth.
The three types of egotism (ahankara) are technically known as vaikarika, taijasa and tamasa. The mahat-tattva is situated within the heart, or chitta, and the predominating Deity of the mahat-tattva is Lord Vasudeva.
In Ilāvṛta-varṣa, Lord Śiva is always encircled by ten billion maidservants of goddess Durgā, who minister to him. The quadruple expansion of the Supreme Lord is composed of Vāsudeva, Pradyumna, Aniruddha and Saṅkarṣaṇa. Saṅkarṣaṇa, the fourth expansion, is certainly transcendental, but because his activities of destruction in the material world are in the mode of ignorance, He is known as tāmasī, the Lord's form in the mode of ignorance. Lord Śiva knows that Saṅkarṣaṇa is the original cause of his own existence, and thus he always meditates upon Him in trance by chanting the following mantra.
The mahat-tattva is transformed into three divisions: Vaikarika, egotism in goodness (sattvika-ahankara), from which the eleventh sense organ, the mind, is manifest and whose predominating Deity is Aniruddha.
Taijasa, or egotism in passion (rajasa-ahankara), from which the senses and intelligence are manifest and whose predominating Deity is Lord Pradyumna.
Tamasa, or egotism in ignorance, from which sound vibration (shabda-tanmatra) expands. From the sound vibration, the sky (akasha) is manifest and, the senses, beginning with the ear, are also manifest.
Regarding the 4 weapon combination we have the 24 Vaikuntha planets in the spiritual sky, wherein each one of them reside as Lords. The 24 forms are all tattvas with a name, which are deified and worshipped in the form of Shaligrama. Each Shaligrama has a particular deity and a method of worship.
Its mentioned in the Garuda Purana.
Chapter XLV - Characteristic marks of Shalagrama Stones (Shaligram)
Hari said:— As a side issue of this discourse I shall, describe the characteristic marks of Shalagrama (Shaligram), by touching which stone, one has the sins, accumulated in a koti of births, dissipated.
The holder of conch-shell, discus, club and lotus, known as Keshava, is [also called] Gadadhara [the holder of mace]. The Lord Narayana is also the holder of lotus, the bow Kaumodaki, discus and conch-shell. He is the holder of discus, conch-shell, lotus and club and is also named Madhava and Shri Gadadhara. Govinda, Gadadhara, the holder of club, lotus, conch-shell and discus is adorable.
Salutation unto thee of the form of Vishnu, the holder of lotus and the conch-shell, unto thee of the form of Madhusudana, the holder of conch-shell, lotus, club and discus. Salutation unto that form of thine which holds the club, conch-shell and lotus, unto that form, which is of three foot steps, unto that form which holds the bow Kaumodaki, lotus and conch-shell and unto thy dwarfish form Vamana. Salutation unto thee the holder of conch-shell, lotus, discus and club. Salutation unto that form which bears the mystic mark of Srivatsa, unto Hrishikesha, and the wielder of lotus, club, conch-shell and discus. Salutation unto the holder of lotus, discus, club, conch-shell, unto Padmanabha (lotus-navelled), unto Damodara, unto the wielder of conch-shell, discus, club and lotus. Salutation unto Vasudeva, unto the holder of conch-shell, club and lotus. Salutation unto Sankarshana. Salutation unto the form of Pradyumna, the holder of a beautiful conch-shell, a beautiful club and a beautiful lotus. Salutation unto Aniruddha and the wielder of club, conch-shell and lotus. Salutation unto the Purusottama form with lotus, conch-shell, club and discus. Salutation unto the form of Adhokshaja, the holder of club, conch-shell, and lotus. Salutation unto Man-lion form, the holder of lotus, club and conch-shell. Salutation unto the form, Achyuta, the wielder of lotus, conch-shell and club. I will bring here Janarddana with his conch-shell, discus, lotus and club and Upendra with his wheel, club, lotus and conch-shell Salutation.
Salutation unto this form of Hari, holding a beautiful discus, lotus, dub and conch-shell. Salutation unto the form. Srikrishna weilding club, lotus, wheel and conch-shell.
The Shalagrama stone, known as Vasudeva, is of a white colour and possesses two ring-like marks around its mouth. The one, known as Sankarshana, is of a red colour, has two ring-like marks around its mouth or lateral aperture and has the mark of a lotus on its eastern side. The one, known, as Pradyumna, is of a yellow colour, has a small ring-like mark, is of an elongated shape and bears on its surface a large number of impressions like pin-holes. The one, known as Aniruddha, is of a circular shape and blue colour and is marked with three lines around its mouth. The one known as Narayana is of a black colour, contains the mark of a club-like line in its cavity and has the circular mark at the navel or a little raised up. The Narasimha alone is elevated on the breast, is of a tawny colour, and is dotted with fine spots; while a Varaha resembles a Sakti in shape and has two rugged and uneven circular marks. A Sthula stone is of blue colour, has three lines, is of the form of a tortoise and is dotted with marks. The stone, known as Krishna, is round and depressed at the back. Sridhara has the marks of five lines, that of a garland of forest flowers and that of a club. The Vamana stone is of a round and puny size and Sureshvara has a circular mark on its left side. The one, known as Anantaka, has various forms and bears an impression like the hood of a serpent. The Damodara stone is thick, of blue colour, and contains a circular mark of blue colour in the central part of the cavity. Brahma stone has a small mouth and is of thick blue colour. Sushira stone has marks of long lines, while Amvuja stone has the > mark of one circular figure and is thick. A Sthulachakra stone has deep holes and is dotted with dark marks. A Hayagriva stone is of the shape of a goad; while a Kaustava stone possesses marks of five lines. A Vaikuntha stone is of the hue of a gem on the hood of a serpent, of dark colour and bears one circular mark. A Matsya stone is of the shape of a long lotus and has marks of lines at the mouth. A Trivikrama stone bears the mark of a circle on the left side, that of a line on the right side and is of a dark-blue colour.
Salutation unto the holder of club who is situate in Shalagrama (Shaligram) as well as in Dvaraka.
A Lakshmi Narayana stone is of the shape of a Kadamva flower, bears marks of four circles at one mouth, is adorned with the figure of a garland of forest flowers, is marked with golden lines and cow’s hoofs.
The Sudarshana class has got only a single circular mark, while the existence of two characteristics the class Lakshmi Narayana. The Trivikrama class has got three circular marks, the Chaturvyuha class has got four, the Vasudeva class has got five, the Pradyumna class has got six, the Sankarshana class has got seven, the Purusottama class has got eight, the Navavyuha class has got nine, the Dashavatara class has got ten, the Aniruddha class eleven, and the Dvadashatma class has twelve circular marks. The Ananta class bears more marks than twelve. He, who reads this poem, describing the forms of Vishnu, repairs to the celestial region.