Yes. IME leaving out food for birthdays means it's for everyone. It would be a very petty minded person who would begrudge any colleague (and you are a colleague, little fish or not) a piece of birthday cake. But it won't hurt to ask, people will appreciate your manners.
1. A general practitioner (ordinary family doctor) is the place to start. If it's a more complex issue he might refer to a proctologist (bottom doctor).
2. Depends. Could be creams,dietary advice, freezing, cutting off under local or in bad cases surgery may be required