This question already has an answer here:
First example: When a creature is going to attack me alone and I use Vapor Snag to return it to the other player's hand, is the combat damage prevented?
Second example: If three 1/1 Soldier creatures are going to attack me while Veteran Swordsmith (which gives other Soldier Creatures +1/+0) is on the board and I use Doom Blade to kill the Veteran Swordsmith, do I take 3 damage or 6?
Third example: I control a creature, but couldn't use it to block a sliver my opponent controls because he just put a Shifting Sliver into play. If I use Path to Exile to exile his Shifting Sliver during the blockers declaration phase, does this make me able to block the other sliver?
Also, would this kind of play work in response to triggered/activated abilities? (for example, trying to prevent a triggered/activated ability from resolving by removing the permanent from the battlefield in response to the opponent trying to trigger/activate it)