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?