I am a little new to this too, but from other members' posts this is what I can answer for you so far:
(1) The best tool to use to split a pup is a digging bar. You can find it here:
Truper Tamper Digging Bar - Post Hole Digger - Ace Hardware
When you use it you need to make sure that you angle it such that you get a nice chunk of the corm from the parent plant. A good illustration can be found here:
Google Image Result for http://www.hardytropicals.org/images/blog/banana_pups.jpg
(2) A banana plant fruits only once and then it dies.
(3) You should not worry too much about fertilizer until it gets warmer. Lay off the watering too.