Is there any restrictions for using hosted Counters? Agenda counters and the like?
Are they a paid ability? Can you use them at any time in the flowchart (in the back of the rulebook) where the little green arrow is pictured?
Can, say, if you are the corp, can you trigger them in the Runner's turn?
Take Data Raven as a basic example, can you spend hosted Counters on it at 'any' time to give the Runner a tag? What are the restrictions?
Anything written in the form Cost: Effect is a paid ability and can be used at any point in the timing chart showing a green arrow, on either player's turn. The only restrictions are any printed on the card (such as "you may only use this during a run") or if the cost includes one or more clicks, in which case the ability is an Action and can be performed as an Action on your turn only.
So yes, if Data Raven gets a counter, it can be spent to give to the runner at any point in the timing chart with a green arrow - so at almost any point. It's very powerful, and that's why most players will never let Data Raven get a counter unless they are already swimming in tags that they have no intention to clear (called 'going tag-me').