Various questions about Karametra's Favor
1) Karamertra's Favor doesn't tap, it gives an ability to the creature it is enchanting allowing it to tap for mana. This mana works just like any other kind of mana and empties from your mana pool at the end of the step or phase.

2) You can use the ability anytime you are able to tap the creature, so you can do it with blocking creatures and attacking creatures with vigilance.

3) If the enchanted creature is tapped to attack you are not able to use the ability granted by Karametra's Favor. You do not need to tap in order to block so you can still tap the creature for the granted ability after it has been declared as a blocker.