Girls going to universities

Q: I am a 23 years old girl studying in a medical university. I wanted to ask that can I continue my studies in the medical university? Is it allowed
in Islam to get education? The university is co-education but I don't talk to any male. I go to university in proper parda so can I continue it? I have heard that a female cannot go out of home without a mahram. As I don't have a brother, or a husband nor can my father go with me so in such a case is it permitted in Islam to go?

Animal euthanasia

Q: I'm working in animal an animal hospital and  sometimes we got patients like horses in very bad conditions in pain so we use a drug that can kill in seconds. Is there anywhere in quran or hadith or shariya that we shouldn't kill these kind of patients that are in pain?

Sada-e-sahw

Q: I have a stressful problem with waswasah about "two prostrations of forgetfulness" like sometimes, I try to refrain from doing the "two prostrations of forgetfulness" when I feel it isn't obligatory to do it, sometimes I stop and sit about whether it's necessary or not - I probably do averagely two "two prostrations of forgetfulness" a day and that is a lot to me, and other times wondering whether I need to do the "two prostrations of forgetfulness" or not! Do I need to do the "two prostrations of forgetfulness" if I accidently elongate a letter or if I don't elongate when you should accidently? Question: What mistakes are acceptable for the salah to be valid without the obligation of doing "two prostrations of forgetfulness"?