Do opposite counters coexist or cancel each other out?
Zero counters. They cancel each other out.


  121.3. 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 as a state-based
  action, where N is the smaller of the number of +1/+1 and -1/-1
  counters on it.