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"?

Designing logos for haraam products

Q: I am a graphic designer and I want to find out whether it is permissible to design logos for:

1. A bed and breakfast that will also use the logo for the wine and sausages that they produce. (In Italy, there are many bed and breakfasts located in the countryside that also produce and sell typical countryside products such as wines, sausages, etc.)

2. Non-Muslim clothing brands.

3. Can I design business cards and advertising material for businesses that may also sell haraam products?

4. Finally, is it permissible to design logos containing Christian religious symbols (such as the Christian cross, which is found in the flag and coat of arms of many Italian municipalities, or the cross of pharmacies)?