If I cast for example Miscast to counter an instant my opponent controlls, and my opponent pays 3, what happens to Miscast? Is it countered? And what about priority? Do I have a chance to put next Miscast on the stack? It's quite confusing to me.
Miscast resolves normally, it just so happens that part of resolving normally in this case involves a thing that may or may not happen. It's similar to if you cast a spell that says "destroy all creatures" (Such as Day of Judgment) while there are no creatures on the battlefield. It would resolve, but its instructions have no effect on the game - although to be more accurate, it did have an effect; it cost your opponent mana.
When Miscast resolves, that's when your opponent gets the choice of whether to pay X or not. Whether they choose to or not, the spell has already resolved. Priority would go to the active player as it always does after a spell resolves. Whether you are the active player or not, you will still get a chance to cast another Miscast or any other instant before your opponent's original spell resolves.
608.2d If an effect of a spell or ability offers any choices other than choices already made as part of casting the spell, activating the ability, or otherwise putting the spell or ability on the stack, the player announces these while applying the effect.
608.2f If an effect gives a player the option to pay mana, they may activate mana abilities before taking that action.
117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
117.4. If all players pass in succession (that is, if all players pass without taking any actions in between passing), the spell or ability on top of the stack resolves or, if the stack is empty, the phase or step ends.
The choice to pay 3 is made during the resolution of Miscast.
If I cast for example Miscast to counter an instant my opponent controlls, and my opponent pays 3, what happens to Miscast?
It finishes resolving, and goes to your graveyard.
Is it countered?
No, it had it's effect. It cost your opponent mana.
And what about priority?
A spell has just resolved, so whomever's turn it is gains priority, as normal.
Do I have a chance to put next Miscast on the stack?
Yes, you will gain priority again before the target spell resolves