The 'white' poo that you are referring to is called Urates, which pretty much is a snakes form of pee. I feed my yr old female a small adult rat every week, without any problems. The appropriate sized prey for a BP would be a rat that is no larger than the thickest part of their body, so you did good on that one!
Also, it's not uncommon for snakes to start a shed cycle if they've been moved and/or stressed recently. That also goes for the whole pooing/peeing on you for the first few handling times. All that I can tell you is to not handle your BP for about a week so he/she can get settled in, then build up a consistant handling routine every week with short 5minute handling sessions, at the same time every week. Make sure that when you go into his/her enclosure that you open it the same way every time, and come in the cage the same way. I have both of my snakes in a tub rack system, so I can really only come in from one angle, but I make sure to stroke the snakes back (behind the head) right before I pick them up. It's just a conditioning thing for them to learn so they will know that I'm not a predator that's going to scoop them up and eat them.