My neighbor has strep throat and was just over. Should I take ask the doctor for antibiotics to make sure I'm in the clear?

No. The best prevention for not getting strep throat from family members is good hygiene. Without any symptoms, then it is is not neccessary to get tested or start on antibiotics. Also it is important to remember that antibiotics are not without risks or side effects.
Not at all. Strep throat is passed though active or passive contact with infected saliva. Unless you have a close kissing relationship with the neighbor there is no reason to believe you were exposed. About 5-10% of the population are symptom free carriers of strep so you are likely exposed to the same degree daily if you have any public contact. No symptoms=no need to test or treat.