Is Sujud At-Tilawah Allowed at Times Prayer is Forbidden?
Question

When the sun rises and sets, are all optional prayers forbidden, even two Rak`ahs for seeking forgiveness and the two Rak`ahs following Tawaf and the prostration of recitation? What is the evidence for that?
Praise be to Allah.Optional prayers at times when prayer is forbidden
We have already discussed the issue of optional prayers at times when prayer is forbidden. Please see questions no. 306, 8818 and 20013.
Can You Offer Sujud At-Tilawah When Prayer Is Forbidden?
The prostration of recitation is not prayer according to the more correct of the two scholarly opinions. We have already discussed this in questions no. 4913 and 22650.
Based on this, it is permissible to prostrate when reading one of the Quranic verses where prostration is required at times when prayer is not allowed, according to the more correct of the two scholarly views, because that does not come under the same rulings as prayer. Even if we assume that it does come under the same rulings as prayer, it is permissible to do it at times when prayer is otherwise not allowed because it is something that is done for a reason, like the eclipse prayer, and the two Rak`ahs following Tawaf if one has done Tawaf at a time when prayer is not allowed. (Fatawa Al-Lajnah Ad-Da’imah, 7/264)
Can You Offer Salat At-Tawbah When Prayer Is Forbidden?
Your saying, “the two Rak`ahs for seeking forgiveness”; perhaps what you mean is the two Rak`ahs of repentance, which are prescribed when repenting from sin. It was narrated that Abu Bakrah (may Allah be pleased with him) said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say: “There is no man who commits a sin then he gets up, purifies himself and prays two Rak`ahs, then asks Allah to forgive him, but Allah will forgive him.” Then he recited this verse (interpretation of the meaning):
{And those who, when they have committed Fahishah (illegal sexual intercourse) or wronged themselves with evil, remember Allah and ask forgiveness for their sins; — and none can forgive sins but Allah — and do not persist in what (wrong) they have done, while they know.} [Al `Imran 3:135] (Narrated by At-Tirmidhi, 408; Abu Dawud, 1521; Ibn Majah, 1395. Classed as authentic by Al-Albani in Sahih Abu Dawud, 1346)
The two Rak`ahs of repentance constitute a prayer for which there is a reason, so it is permissible to do that at a time when prayer is otherwise not allowed. And Allah knows best.
In other authentic reports there are other descriptions of the two Rak`ahs which expiate for sin. This may be summed up as follows:
-There is no one who does Wudu and does it well (because sins come out from the washed limbs with the water or with the last drop of water)
Part of doing Wudu well is to say Bismillah (In the name of Allah) before starting it, and reciting Dhikr after it, as follows: Ashhadu an la ilaha ill-Allah wahdahu la sharika lah, wa ashhadu anna Muhammadan `abduhu wa rasuluhu. Allahumma aj`alni min Al-tawwabina waj`alni min Al-mutatahharin. Subhanaka Allahumma wa bi hamdika, ashhadu an la ilaha illa anta, astaghfiruka wa atubu ilayk (I bear witness that there is none worthy of worship but Allah alone, with no partner or associate, and I bear witness that Muhammad is His slave and Messenger. O Allah, make me one of those who repent and make me one of the purified. Glory and praise be to You, O Allah, I bear witness that there is no god but You, I seek Your forgiveness and I repent to You). This dhikr recited after Wudu brings a great reward.

Then he gets up and prays two Rak`ahs
In which he does not make any mistake
And does not let his mind wander
And he remembers Allah and focuses with proper humility
Then he asks Allah to forgive him

The outcome will be: that he is forgiven for his previous sins and will deserve Paradise. (Sahih At-Targhib, 1/210-211)
And Allah knows best.