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.

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 for a woman who broke fasting due to a delivery to pay a ransom?

She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.

What is the ruling on the cessation of blood after (40) days from delivery, but later continued sporadically during two days of Ramadan?

Once postpartum bleeding (Nifas) ceases, and the woman is certain that it won`t reoccur, then she becomes ritually pure and so she is free to make Ghusl (purificatory bath), pray, and fast. If the bleeding reoccurs before fifteen days from its cessation, and before the end of (60) days after delivery, then the ruling on postpartum bleeding is effective, and her fasting and prayer are null and void, thus she must make up the fasting that she missed and not the prayer during those particular days.