Does Istigfar (Asking Allah for forgiveness) between the first and the second part of Friday prayer`s sermon render the prayer invalid?

Talking during Friday prayer`s sermon is disliked, but there is no harm in offering Istigfar between the two parts of the sermon, and it doesn`t invalidate the prayer as it is actually from Sunnah. And Allah Knows Best.

Am I permitted to perform more than a prayer within single ablution?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

You are permitted to pray as many as you could within one ablution and to recite Quran so long as your ablution isn't nullified. And Allah Knows Best.

Is it permissible to slaughter one animal as a sacrifice and an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?

It is impermissible to do so since each of them is slaughtered for a different reason.

What is the ruling on someone who vowed to fast for two consecutive months, but failed to do so?

He is obliged to fulfill his vow by fasting for two consecutive months, but if he fails to do so then, it remains a debt on him until he either fulfills his vow, or dies. In case of his death, his guardian is supposed to fast on his behalf, or donate food from his estate in return for fasting, but if the guardian fails to do so then, it is all in the Hands of Allah.