Can I deal combat damage with a 1/1 Walking Ballista and deal damage with its ability?
No, you cannot.  Going to the graveyard (due to damage) is a State-Based action that happens before either player gets priority.  Without priority, you can't activate Ballista's ability. 


  704.3. Whenever a player would get priority (see rule 116, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. If any state-based actions are performed as a result of a check, the check is repeated; otherwise all triggered abilities that are waiting to be put on the stack are put on the stack, then the check is repeated. Once no more state-based actions have been performed as the result of a check and no triggered abilities are waiting to be put on the stack, the appropriate player gets priority.