Let's first start with the following verse from Mahabharata itself.:
“yo ‘sṛjad dakṣiṇād aṅgād brahmāṇaṃ lokasaṃbhavam | vāmapārśvāt tathā viṣṇuṃ lokarakṣārtham īśvaraḥ | yugānte caiva saṃprāpte rudram aṅgāt sṛjat prabhuḥ |” (MBH 13:14:183).
"Thou art he that hadst created from thy right side the Grandsire Brahma, the Creator of all things. Thou art he that hadst created from thy left side Vishnu for protecting the Creation. Thou art that puissant Lord who didst create Rudra (the destroyer deity Kalagni-Rudra) when the end of the Yuga came and when the Creation was once more to be dissolved”.
So, yes, Bhagawan Shiva creates Lord Brahma from his Right, Lord Vishnu from his left and Lord Rudra from his heart.
They are being created to perform different activities.
Padma Purana Patala Khanda Chapter 108.
Yes, they are born from his Desire (Kama) or you can also call it as ahamkara and plus also for sport (lila).
Lord Rudra is called as "Kâma" or Kameshwara because he himself is 'Primodial desire'. The whole universe sprung because of the primodial desire. In the begining, there was neither existance nor non-existance, neither space nor time. There was no second being. Shiva(Brahman) alone breathed without wind. This primodial action is called as "Coming into Existance". This is possible only because of Desire. hence Lord Shiva is called "Kâma" .
Rig veda (Nasadiya Suktam) 10:129:4 says.:
" kāmas tad agre sam avartatādhi manaso retah prathamam yad āsit".
Kâma rose in the beginning, Kâma, the primal seed and germ of Spirit.
The Whole creation aroused because of union of Shiva and Shakti.
No, they are not Jivas they are Brahman themselves but they are saguna brahman viz with forms.
This Chapter of Sri Shiva Purana talks about how Brahma, Vishnu, Rudra, Ishwara and Shiva are the form of Omkara (Pranava). Bhagawan Shiva further states that Omkara is his word which is an indicator and the indicated is him only. So, Shiva = Pranava. Where Brahma (A) is creation, Vishnu (U) is Preservation, Rudra (M) is Destruction, Ishwara (Nada/Sound) is concealing/Maya and Shiva (Bindu/full stop) is Moksha. Its basically Pancha Brahman Form of Lord Shiva viz Sadyojata, VamaDeva, Aghora, Tat-Purusha, Ishana.
"nama staraya cha" (Yajurveda Sri Rudram Anuvaka-8).
"Salutations to Him (Rudra) who is the TARAKA (Pravana mantra - Om)".
2.3 It is advised that the five gods Brahma, Vishnu, Rudra, Isvara and Shiva should be worshipped in the form of pranava [Aa + Uu + Ma + sound + Bindu (full stop)].
This Chapter of Sri Shiva Purana tells why Bhagawan Shiva took the saguna Rudra avatar from Lord Brahma because of the request from Lord Brahma and Lord Vishnu. He further says that Brahma, Vishnu, Rudra are his own forms which are created from Ahamkara so, the have gunas like Raj, Sat, Tam, while Brahma and Vishnu are of Prakriti tatva, Rudra is totally Purusha Tattva and his complete incarnation. Like Gold and Ornament have different names but same elements same as for clay and clay models all 3 are his own forms.
It can be seen as follows.:
Rudra (Purusha) + Vishnu (Prakriti) gives birth to Brahma (Son/Hiranyagarbha) so he came out of Vishnu from navel stalk with lotus. Which signifies the umbilical cord of a baby.
Vishnu = Uma. As both are present in the left side of Shiva. When Uma becomes male she becomes Vishnu and when Vishnu becomes female he becomes Uma.
Hiranyagarbha means Golden egg/womb/featus/son/child.
Purusha + Prakriti = Universe (Brahma-Anda/Hiranyagarbha).
Sri Shiva Purana 3:11:40.:
प्रकृतिस्त्वं पुमान्रुद्रस्त्वयि वीर्य्यं समाहितम् ।
त्वन्नाभिपङ्कजाज्जातः पञ्च वक्त्रः पितामहः ॥ ४० ॥
40. You (Vishnu) are Prakṛti and Rudra is Puruṣa. The seed is deposited in you. The five-faced Brahmā was born of your umbilical lotus.
Same thing is represented by Adi Shankara said in Shivananda Lahari as follows.:
"Ardha vapusha baryatwam, Gonithwam sakkhitha..." (Shivananda Lahari Para-82).
"Lord Vishnu occupied half your body, Became transformed to be your wife (PARVATI), Became a boar to search for you (LINGA), Became your lady friend to serve the nectar (MOHINI)."
2.2 Brahma, Vishnu, Rudra and Indra are creating all beings, all organs and all karanas. They are also capable of controlling them. But Lord Shiva exists in between them like sky (Akasa/Atma), and is permanently stable.
Again, The both Nirguna and Saguna forms of Bhagawan Shiva is beautiful explained by Lord Vishnu to Maa Lakshmi in Padma Purana.
In this Verse Lord Vishnu Says to Mata Lakshmi that his Chaturbhuja Tamala Varna form is an illusion just for the purpose of 3 duties. His real form is that of Bhagawan Shiva.
śrībhagavānuvāca | māyāmayamidaṃ devi vapurme na tu tātvikam | sṛṣṭisthityopasaṃhārakriyājālopabṛṃhitam || 16 ato’nyadātmanorūpaṃ dvaitādvaitavivarjitam | bhāvābhāvavinirmuktamādyaṃtarahitaṃ priye ||17 śuddhasaṃvitprabhālābhaṃ parānaṃdaikasuṃdaram | rūpamaiśvaramātmaikyagamyaṃ gītāsu kīrtitam ||” (Padma Purana:UK:175:16-18).
“[Lord Vishnu said]: O goddess (Lakshmi), this my body is illusory and not real, and is augmented with the mass of the acts of creation, maintenance and withdrawal. O dear one, the nature of the self is different form this. It is without duality and unity. It is free from existence and non-existence; and without beginning or end. It is pure consciousness, has acquired luster, is beautiful due to great joy, is the form of the Ishvara (Shiva), can be known only through the unity with the self, and this is what is told in the Gita”.
Here Lord Vishnu describes how Thete are 2 forms of Bhagawan Shiva, one Nirguna, Nirakar and other is Panchavakra and Maheshwara.
“īśvara uvāca | śriyaḥ śrutvā vacoyuktamitihāsapuraḥsaram | ātmānugāminīṃ dṛṣṭiṃ gītāṃ bodhitavānprabhuḥ ||20 ahamātmā pareśāni parāparavibhedataḥ | dvidhā tataḥ paraḥ sākṣī nirguṇo niṣkalaḥ śivaḥ ||21 aparaḥ paṃcavaktro’haṃ dvidhā tasyāpi saṃsthitiḥ | śabdārthabhedato vācyo yathātmāhaṃ maheśvaraḥ ||22 gītānāṃ vākyarūpeṇa yannirucchidyate dṛḍhaḥ | madīyapāśabaṃdho’yaṃ saṃsāraviṣayātmakaḥ ||23 yadābhyāsaparādhīnau paṃcavaktramaheśvarau | iti tasya vacaḥ śrutvā gītāsāramahodadheḥ ||24 idaṃ paravibhedena budhyate bhavabhīrubhiḥ | tamapṛcchadidaṃ lakṣmīraṃgapratyaṃgasaṃsthitam ||25 māhātmyaṃ setihāsaṃ ca sarvaṃ tasyai nyavedayat |26a” (Padma Purana:UK:175:20-26a).
“[Lord Shiva (Ishwara) said]: Having heard the proper words of Lakshmi, the lord told her the Gita, the vision leading one towards the self, along with an historical account. “O great lordess, I am the ‘self’ differentiated in two ways, – as higher and lower. The higher is a witness, attributeless, partless, Shiva. As the lower one, I am ‘panchavaktra (of five faces)’ it also remains in two forms. I, Maheshvara, am the self, am to be explained through the distinction in words and meanings, as, through the words of the Gita, my strong noose of the nature of objects in the worldly existence, is completely cut off; since (realization of my two forms) ‘Panchavaktra’ and ‘Maheshvara’, are dependent on its study“. Hearing these words of him (which were) the great ocean of essence of the Gita, those afraid of the worldly existence know it through the difference between this and that. Lakshmi asked thus to him. He told her about the Mahatmayam (greatness) along with the corresponding historical accounts (stories) in its major and minor details.”.
Then he goes on saying that the 18 chapters of Bhagwat Gita are the body parts of Panchavakra Shiva himself.
“śṛṇu suśroṇi vakṣyāmi gītāsu sthitimātmanaḥ ||26 vaktrāṇi paṃca jānīhi paṃcādhyāyānanukramāt | daśādhyāyā bhujāścaika udaraṃ dvau padāṃbuje ||27 evamaṣṭādaśādhyāyā vāṅmayī mūrtiraiśvarī | vijñeyā jñānamātreṇa mahāpātakanāśinī ||28 atodhyāyaṃ tadardhaṃ vā ślokamarddhaṃ tadardhakam | abhyasyati sumedhā yaḥ suśarmeva samucyate ||” (Padma Purana:UK:175:26b-29).
“[Lord Vishnu said]: O you, beautiful lady, I’ll tell you about my firm existence in the Gita. The five chapters are the five faces in order. The (next) ten chapters are the (ten) hands; one is the belly, and the (remaining) two are the lotus like feets. Thus the eighteen chapters are the divine forms of words. It, destroying great sins, should be known through knowledge only. Therefore, he, the very intelligent one, who studies a chapter, or half of it, or a verse of a half-verse is liberated like ‘Susharma’”.
Whom do the Tridevas worships.:
2.1 The pranava (the sound of Aum) makes all the souls to bow before it. It is the one and only one which has to be meditated upon as the four Vedas and the birth place of all devas. One who meditates like that goes away from all sorrows and fears and gets the power to protect all others who approach him. It is because of this meditation only that Lord Vishnu who is spread every where, wins over all others. It is because Lord Brahma controlled all his organs and meditated upon it, he attained the position of the creator. Even Lord Vishnu , parks his mind in the sound (Aum) of the place of Paramathma (ultimate soul) and meditates upon Eeshana, who is most proper to be worshipped. All this is only proper in case of Eeshana.
In Koti Rudra Samhita Shiva Purana chapter 42, Suta states.:..
सर्वे रुद्रं भजन्त्येव रुद्रः कंचिद् भजेन्न हि ।
स्वात्मना भक्तवात्सल्याद् भजत्येव कदाचन ।।
All worship Rudra but Rudra worships none. But for the sake of devotees sometimes he worships/meditates upon His own Nirguna Self (Pranava/Linga)."
I hope this clarifies all your queries. Prd..