How to properly play activated abilities of twilight drover and other creatures?
1: Activated abilities always have the pattern "cost:effect", and as long as you can pay the costs, you can activate it as often as you like. There are a few exceptions where you have a limited number of activations per turn, but those will always be printed on the card. Note that the TAP symbol is a cost like any other, and if you manage to untap your permanent, you can use that ability again without issue.

2: You can activate non-mana, non-Planeswalker abilities at instant speed. Again, exceptions are printed on the card.

3: There are no such restrictions, except for Planeswalker abilities.

4: Yes, but very few cards do that, like Stifle. An ability is not a spell, so cards that counter spells do not work on abilities.