Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
What is the ruling on a person who broke the fast a day in Ramadan due to hardship?
He/she is obliged to make up that day.
Should a pregnant woman who broke fast because of pregnancy make it up, and is a ransom due on her?
The pregnant and the suckling, if they fear for their health, may break their fast and make up for it, and no ransom is due on them. However, if they broke fast in fear for the fetus and the baby, then they are obliged to make up for it, and pay the ransom which is feeding a needy person for each of the missed fasting days. And Allah Knows Best.
Is Aqeeqah due on the miscarried embryo?
No Aqeeqah is due on the miscarried embryo.