Is it obligatory to renew the intention for prayer after the intention for ablution for the obligatory prayer and before the opening Takbir, and do the two rak'ahs of repentance require an intention when performing ablution?

If the intention for ablution (wudu') for prayer is not separated from the prayer by a long interval, then there is no need to renew the intention when saying the takbir, as the four madhhabs permit advancing the intention slightly before the takbirat al-ihram (definition). As for the prayer of repentance (salat al-tawbah (definition)), whoever desires its reward should intend repentance before the prayer. If one prays without the intention of repentance and then seeks forgiveness, the objective is achieved, but without the reward associated with intending repentance before the prayer.