Rudra is both Born and Unborn. As I discuss in my answer here,
Rudra means who cries, who makes other cry and who removes the cry of Samsara.
There may be confusion regarding born and unborn nature of Rudra with the Smriti. However Vedas themselves attest both Born and Unborn Rudra:
He is the ParaBrahman or self of all. He is represented in formless Linga form or Saguna Upasakas also call him SadaShiva. He is unborn and eternal. He is without beginning and end.
Vedas themselves declare this Rudra who existed before Prajapati and who caused Prajapati to do action:
व्रात्य आसीदीयमान एव स प्रजापतिं समैरयत् ।।
स प्रजापति: सुवर्णमात्मन्नपश्यत्तत्प्राजनयत् ।।
तदेकमभवत्तल्ललाममभवत्तन्महदभवत्तज्जेष्ठमभवत्तद्ब्रह्माभवत्तत्तपोऽभवत्तत्सत्यमभवत्तेन प्राजायत ।।
सोऽवर्धत स महानभवत्स महादेवोऽभवत् ।।
स देवानामीशां पर्यैत्स ईशानोऽभवत् ।।
स एकव्रात्योऽभवत्स धनुरादत्त तदेवेन्द्रधनुः ।।
नीलमस्योदरं लोहितं पृष्ठम् ।।
नीलेनैवाप्रियं भ्रातृव्यं प्रोर्णोति लोहितेन द्विषन्तं विध्यतीति ब्रह्मवादिनो वदन्ति ।। [Atharva Veda 15:1]
There was Vratya . He roused Prajapati to action. Prajapati beheld gold in himself and engendered it. That became unique, that became distinguished, that became great, that became excellent, that became Brahman, that became Tapas, that became Truth: through that he (Vratya) was born. He grew, he became great, he became Mahadeva. He gained the lordship of the Gods. He became Ishana. He became Eka Vratya. He held a bow, even that Bow of Indra. His belly is dark-blue, his back is red (Nila Lohita). With dark-blue he envelops a detested rival, with red he pierces the man who hates him: so the knowers of Brahman say.
As clearly evident from above Shruti passage, the one who caused Prajapati [Brahma] to action is the same who was born from him and was known by the name of Mahadeva, Ishana, Nilalohita, etc.. ie. He himself was the cause of his birth and he was controlling Prajapati himself before his birth itself ie. He is unborn. So, anyone who think Shiva is only son of Brahma directly contradict with this Shruti passage. Thus they are wrong. That's why Svetasvatara Upanishad declare:
यो देवानां प्रभवश्रोच्वश्रच् विश्वाधिपो रुद्रो महर्षि ।
हिरण्यगर्भं जनयामानस पूर्वं स नो बुद्धया शुभया संयुनक्तु।।
He, the omniscient Rudra, the creator of the gods and the bestower of their powers, the support of the universe, He who, in the beginning, gave birth to Hiranyagarbha−may He endow us with clear intellect!
He is that SadaShiva who is the cause of all causes. He himself is in the form of Brahma, Vishnu and Mahesh:
For eg He himself became Vishnu:
स यद्ध्रुवां दिशमनु व्यचलद्विष्णुर्भूत्वानुव्य चलद्विराजमन्नादिं कृत्वा ।। विराजान्नाद्यान्नमत्ति य एवं वेद।। [Atharvaveda 15.14]
He (Vratya/ Mahadeva/ Ishana ), when he went away to the stedfast region, went away having become Vishnu and having made Virāj an eater of food”.
That's why SriRudram of YajurVeda declares:
नमो गिरिशाय च शिपिविष्टाय च मेढुष्टमाय च
Salutations to him who is in the form of Girisha (Shiva), Sipivishta (Vishnu) and Medhustama (Brahma).
This is also illustrated in Smriti passages like:
For eg. Mahabharat Anushasana Parva chapter 14 states:
योऽसृजद्दक्षिणादङ्गाद्बर्ह्मणं लोकसंभवम् ।
वामपाश्र्वातथा विष्णु लोकरक्षार्थमीश्वर ।।
युगान्ते चैव संप्राप्ते रुद्रमङ्गात्सृजत्प्रभु ।।
You create Brahma from Right side and make the creation of world possible. From the Left side you emerge as Vishnu for protection of entire world. You create Rudra for the end of the world from your body.
Also Srimad Bhagvatam 8.7.23 states the same thing:
गुणमय्या स्वशक्तयास्य सर्गस्थित्यत्ययान्विभो ।
धत्से यदा स्वद्धग्भूमन्ब्रह्मविष्णुशिवाभिधाम् ।।
O Lord [SadaShiva], you are self-effulgent and supreme. You create this material world by your personal energy, and you assume the names Brahmā, Viṣhṇu and Maheśvara when you act in creation, maintenance and annihilation.
Padma Purana also states the same thing in Patala Khanda chapter 108:
य एकः शाश्वतोदेवोब्रह्मवंद्यः सदाशिवः ।
त्रिलोचनो गुणाधारोगुणातीतोऽक्षरोव्ययः ।।
He who is that eternal god SadaShiva, who is saluted by Brahma, who has three eyes, who is the prop of virtues, who is beyond qualities, who is unchangeable and immutable. O dear one, having divided himself and the region there, he created on his right side the son, viz. Brahma and Hari from his left side. At the back side he created Mahesh; thus the mighty one created three sons. As soon as they were born they became the three gods—Brahma, Vishnu and Maheswara.
Thus the Unborn SadaShiva who is the cause of all causes and who is not son of Brahma but he himself is the Generator of Brahma is attested in both Shruti and Smriti.
But this SadaShiva is also called by the name Rudra in Vedas. There are also Born Rudras described in Vedas. So, one has to look the context to know whether Born Rudra or Unborn Rudra (SadaShiva) is described.
SadaShiva is described of having 5 heads. His five heads are
Sadyojata, Vamadeva, Aghora, Tatpurush, Ishana.
These 5 heads are described in Taittariya Aranyaka of YajurVeda from 10.17 to 10.21. The Upper facing head Ishana is the Ishwara of the world at all times. Hence, Taittariya Aranyaka of YajurVeda [10.21] declares:
ईशान सर्वविद्यानां ईश्वर सर्वभूतानां ब्रह्माधिपति ब्रह्मणोधिपति ब्रह्मा शिवो मे अस्तु सदाशिवोम् ।।
May the Supreme Ishana who is the ruler of all knowledge, controller of all created beings, the preserver of the Vedas and the one overlord of Hiranyagarbha, be benign to me. I am the SadaShiva described thus and denoted by Pranava.
Srimad Bhagvatam also describes form of SadaShiva as:
मुखानि पञ्चोपनिषदस्यवेश यैस्त्रिंशदष्टोत्तरमन्त्रवर्ग ।
यतच्छिवाख्यं परमात्मतत्त्वं देव स्वयंज्योतिरवास्थितिस्ते।।
O lord, the five important Vedic mantras are represented by your five faces, from which the thirty-eight most celebrated Vedic mantras have been generated. Your Lordship, being celebrated as Lord Śhiva, is self-illuminated. You are directly situated as the supreme truth, known as Paramātmā.
Thus, SadaShiva or Linga form Parameshwara is the cause of all causes. He is the Unborn Rudra described in Svetasvatara Upanishad and other Shruti passages like:
देवानां हृदयभ्यो। [YajurVeda SriRudram]
who is on the heart of Devas (as their self).
भवस्य हेत्यै जगताम् पतये नमः । [Sri Rudram]
Salutations to him who being the cause is the Lord of the Universe.
मृत्युर्मुक्षिय मामृतात् ।। [RigVeda 7.59.12] who Gives moksha and Immortality.
नमः सोमाय च रुद्राय च [SriRudram YajurVeda] Salutation to him who is with Uma and who destroys the misery of Samsara.
एको हि रुद्रो न द्वितियाय [Svetasvatara Upa.] Rudra is truly one and only one without second.
त्वं ज्ञात्वामृता भवन्ती । [Svetasvatara Upa.] knowing you one becomes Immortal.
Thus SadaShiva is the Unborn and eternal Rudra who is the cause of all causes.. Shankara who is sitting in Kailash Parvata with Parvati is just physical manifestation/incarnation of SadaShiva.
There are various types of born Rudras as I discuss in my answer here. Vedas themselves attest born Rudras also. The story of Ashtamurti manifestation of Rudras is described in detail in Kaushtiki Brahmana and Satapatha Brahmana. He cried just after he was born. That's why Brahma named him Rudra (rud=cry) as described in this chapter of Satapatha Brahmana.
He said to him, 'Thou art Rudra' And because he gave him that name, Agni became suchlike (or, that form), for Rudra is Agni: because he cried (rud) therefore he is Rudra. He said, 'Surely, I am mightier than that: give me yet a name!'
Brahma goes on naming his 8 forms as:
Rudra, Bhava, Sarva, Ugra, Bhima, Pasupati, Mahadeva, Ishana
There are also Ekadash Rudras who are incarnation of Lord Shiva:
They Rule on Prana and Mind of Body as described in Brihadaranyaka Upanishad:
कतमे रुद्रा इतिः दशेमे पुरुषे प्राणाः आत्मैकादशः ते यदास्माच्छरीरान्मर्त्यादुत्क्रामन्त्यथ रोदयन्ति, तद्यद्रोदयन्ति तस्माद्रुद्रा इति ।।
‘Which are the Rudras ?’ ‘The ten pranas in the human body, with the mind as the eleventh. When they depart from this mortal body, they make one’s relatives weep. Because they then make them weep, therefore they are called Rudras.’
Rudras are given birth by other also for eg:
Srimad Bhagvatam 6.6 states:
17-18. Sarupa, the wife of Bhuta, gave birth to the ten million Rudras, of whom the eleven principle Rudras were Raivata, Aja, Bhava, Bhima, Vama, Ugra, Vrsakapi, Ajaikapat, Ahirbradhna, Bahurupa
and Mahan. Their associates, the ghosts and goblins, who are very fearful, were born of the other wife of Bhuta.
Rudras are also born from Narayana also:
Narayana Upanishad says: "Narayana Rudro Jayet" or Rudra was born from Narayana.
In Mahabharat Shanti Parva chapter 342 also Krishna says:
He should be known as one that has sprung from my wrath.
Srimad Bhagvatam 5.25.3 also tells Rudras manifesting from wrath of Sesha Naag:
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 Sankarsana, is the embodiment of the eleven Rudras, or incarnations of Lord Siva. He appears in order to devastate the entire creation.
Thus as we see from above Rudras get birth from Brahma, Vishnu, Sarupa, Shiva, Sesha, etc.. in different conditions. Also the name of Ekadash Rudras change frequently in Puranas due to KalpaBheda.
Incarnations of Lord Shiva:
There are various incarnations of Lord Shiva the most famous being Ashtamurti Manifestation. He manifest as son of Brahma in 8 different forms and enters the creation as:
Rudra, Bhava, Sarva, Ugra, Bhima, Pasupati, Mahadeva, Ishana
Ekadash Rudras are also his incarnations. Their name frequently changes in Puranas due to KalpaBheda.
Besides Ashtamurti incarnation and Ekadash Rudra incarnation there are also other incarnations in various Kalpas and Mahayugas:
This excerpt from ShatRudra Samhita of Shiva Purana describes incarnation in various Kalpas like:
Sadyojata incarnation in Svetalohita Kalpa. Vamadeva incarnation in Rakta Kalpa. Aghora incarnation in Pitavasas Kalpa. Ishana incarnation in Visvarupa Kalpa.
The same excerpt in Chapter 5 also lists various incarnation taking place in various Mahayugas. These incarnations were named as:
Bali, Gautama, Vedasira, Gokarna, Guhavasin, Sikhandin, Jati, Malin, Doruka, Bhava, Halayudha, Sveta, Sulin, Mundisvara, Sahisnu, Somasarma, Lakulin.
Death of Rudra:
There are not any scripture which mention Rudras died at particular time. Personally I think at the end of Kalpa or Mahakalpa they merge back to Lord Shiva.
Also there are also not any scripture which state Shiva/Rudra is a title and one can achieve this position by doing this and this.