Unless stated otherwise, activated abilities can be activated whenever you could cast an instant. Since Ballista's does not state otherwise, you can activate its ability on your opponent's turn if you want.
Follow up question - Am I limited to paying 4 mana per turn to add the +1/+1, or can I pay 8 and get +2/+2?
You can use the ability as many times as you can pay the cost unless a limit is explicitly stated on the card or in its ability's reminder text. Only planeswalkers have innate activation limits.
Can counters be removed for damage during your opponents turn?
Unless stated otherwise, activated abilities can be activated whenever you could cast an instant. Since Ballista's does not state otherwise, you can activate its ability on your opponent's turn if you want.
Follow up question - Am I limited to paying 4 mana per turn to add the +1/+1, or can I pay 8 and get +2/+2?
You can use the ability as many times as you can pay the cost unless a limit is explicitly stated on the card or in its ability's reminder text. Only planeswalkers have innate activation limits.