Yes, in each creation Lord Rudra (who is one among Trinity) manifests himself through forehead of Brahma. He merges back to Shiva after finite time.
Rudra and Shiva aren't exactly same. There is a subtle difference. Rudra is the Saguna manifestation of Nirguna Shiva. I also suggest to see my answer here:
Is Shiva or Rudra born and is he still alive? Was he killed or merged with supreme brahman at any point of time?
Sri Rudram of YajurVeda which describes hundreds of types of Rudra also states
विरुपेभ्यो नमो विश्वरुपेभ्यश्चभो नमो नमो
Salutations to him who is formless and who is in Universal form.
As I discuss in my answer here there are various types of Rudra like who cry, who makes other cry and who removes cry of Samsara.
Generally Shiva refers to the formless aspect and Rudra denotes his formed Saguna aspect although both names are used synonymously.
Here is description from Koti Rudra Samhita Shiva Purana Chapter 42: Suta states the below verses there:
What is Shiva ?
ब्रह्मादितृणपर्यन्तं यत्किञ्चिद्दृश्यते त्विह ।
तत्सर्वं शिव एवास्ति मिथ्या नानात्व कल्पना ।।
Right from Brahma to the twig of leaf, ie . Everything in this Universe is Shiva himself. The concept of manifestation is not real.
स एव शड़्कर साक्षात्सर्वानुग्रहकारक ।
कर्ता भर्ता च हर्ता च साक्षी निर्गुण एव सः ।।
He himself is the Shankara who graces all. He is the creator protector and destroyer. He is the cosmic witness and Nirguna.
Why did Formless and Nirguna Shiva assumed form?
उभयोर्वादशमने तद्रूपंदर्शितं बुधां ।
महादेवेति विख्यातं निर्गुणेन शिवेन हि ।।
Oh Brahmanas in order to remove controversy between Brahma and Vishnu, the form in which Shiva appeared, came to be known as Mahadeva.
He also at that time tells that he shall manifest through forehead of Lord Brahma:
तेन प्रोक्तमहं शम्भुर्भविष्यामि कभालतः ।
रुद्रो नाम स विख्यातो लोकानुग्रहकारक ।।
He said, “I shall appear from the forehead of Brahma in the form of Sambhu.' The same Sambhu, the bestower of welfare came to be known as Rudra in due course of time.
ध्यानर्थं चैव सर्वेषामरुपो रुपवानभूत् ।
स एव च शिवः साक्षाद्भक्तवात्सल्यकारकः ।।
Since that date for the meditation of the devotees of the lord, Shiva, took the Sakara (with form) for the benefit of his devotees. He therefore likes the devotees of Shiva.
These above two verses clearly show difference between Nirguna Shiva and Saguna Rudra.
What is the relation between Shiva and Rudra?
शिवे त्रिगुणसम्भिन्ने रुद्रे तु गुणधामनि ।
वस्तुतो न हि भेदोऽस्ति स्वर्णे तद्भूषणो यथा ।।
There is absolutely no difference between Shiva who is beyond the three gunas (satva, rajas and tamas) and Rudra, possessing all the gunas, in the same way as there is no difference between the gold and the gold ornaments.
समानरुपकर्माणौ समभक्तगतिप्रदौ ।
समानाखिलसंसेव्यौ नानालीलाविहारिणौ।।
Both of them perform their activities similarly, bestow salvation to their devotees similarly. Both of them are served by the people equally well and they roam about indulging in several types of divine sports.
Is Rudra who is born from forehead of Brahma eternal ?
No, as he has birth/manifestation he must merge back to his original source.
अन्ये च ये समुत्पन्ना यथानुक्रमतो लयम् ।
यान्ति नैव तथा रुद्रः शिवे रुद्र विलियते ।।
Others which have origin finally dissolve as per their process. However Rudra merges back to Shiva himself.
Who is married with Sati and Parvati ?
Rudra who came from forehead of Brahma and lives in Kailash married with Sati and Parvati. Actually he married Sati and Parvati many many times.
Brahma while Praying to Durga in Shiva Purana to ask her to incarnate as Sati (which I discuss here) says this:
य पतिस्तव देवेशि ललाटान्मेऽभवत्पुरा ।
शिवो रुद्राख्या योगी स वै कैलासमास्थित ।।
O Goddess of the Devas, Your husband had been born out of my forehead and known as Rudra and is dwelling at Kailash as Shiva.
So it is clear Rudra Avatar of Lord Shiva takes place as cyclic phenomena as described in Vedas. In each period of Brahma, Lord Shiva incarnates several times through his forehead. Some of his incarnation like Rudra Manifestation, AshtaMurti manifestation, Ekadash Rudra manifestation etc..(which I discuss here are described in Vedas itself.)
Other stories also repeat in similar ways. Coincidental circumstances are itself created to make story described in Vedas happen as Vedas are eternal truth themselves.
