Can you activate Remote Revenge if your opponent flips either Old Vindictive Magician or Man-Eater Bug?

Answer 1

No, and Yes. Remote Revenge can only change the target to another legal target that the opponent controls. Old Vindictive Magician cannot target its own controller's monsters, meaning it's impossible for Remote Revenge to work against it. Remote Revenge could not be activated against it.

But Man-Eater Bug has no such restriction, and at the very least, could be forced to target itself for destruction.

Answer 2

No. To be valid for Remote Revenge to be activated, the spell, trap or monster effect has to destroy the target on resolution of that effect. It's not enough for that card to include a destruction effect somewhere in the text. So even with an Equip Spell Card that will destroy the equipped monster on a future turn, like Ekibyo Drakmord, Remote Revenge can't be used against it.

Answer 3

No, Remote Revenge switches the target to 'an appropriate monster your opponent controls'. 'Appropriate' in this case means the activated effect still has to pick a target that is valid. That means it can never be used against a card that specifies the opponent's monsters, there are no legal targets on the activating player's own side of the field.

As a general rule of thumb, ask yourself, could the opponent have used the effect on one of his monsters in the first place. If not, then Remote Revenge can't switch the target. The opponent could use Nobleman of Crossout on one of his own monsters. If he used it on yours, Remote Revenge could switch the target to one of his Set monsters. The opponent could not use Old Vindictive Magician on his own monsters, therefore if he uses it on yours, Remote Revenge cannot switch the target.