Do +1/+1 counters go away with -1/-1 counters?
Yes, +1/+1 counters and -1/-1 counters will cancel each other other. This is done as as state-based action, any time a player is about to receive priority.

704.1. State-based actions are game actions that happen automatically whenever certain conditions (listed below) are met. State-based actions don’t use the stack.
704.5. The state-based actions are as follows:

704.5r If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it, where N is the smaller of the number of +1/+1 and -1/-1 counters on it.