Fog bank

[MementoMori]

If you attack fog bank with a 1/1 does your creature die?

Even if it got more than 0 power somehow, no. Its damage is prevented as well.

If you attack it with any creature does your creature die?

No. Why would it?

And what if your creature has death touch???

Deathtouch requires the creature to deal damage in order to work. So in combat, deathtouch will do nothing, because the damage is prevented.