I can help answer #2 and #3.
#2. I received an NSF fellowship in grad school with 6 months of research experience. The experience probably helped, but getting an NSF is a long shot no matter what. I know of other students with similar qualifications who didn't get it.
#3. The prestige of your grad school matters in some circles but not in others. If there are particular companies that you want to work for, try to find out what their hiring practice is. Some companies place a high value on the school you graduated from, and some don't care at all.