If I get hurt whether that's during BMT or SW prep, I was wondering if that would affect my chances of getting into A&S or if that would disqualify me from the PJ route?
If you get hurt then the severity of your injury will determine your ability to stay in the pipeline. If you have an injury that will take over 60 days to heal then you will most likely be reclassified. If it is under then you have a good chance of being recycled given that you have earned it of course.
After basic training, when will I be able to normally use my phone to video chat with my family and friends?
You will be able to have your phone at some points even through basic, but you will always have it after duty hours at selection. You will have an opportunity to spend time with your family after you graduate basic. But you need to get your mind right, YES talking with family is very important, however you need to not worry about that. Focus on passing and getting through.