Update: Seems that the card in question is The Ultimate Creature of Destruction, targeting any Blue-eyes monster. In that case, my previous answer holds.
So, if my opponent activates a card that makes me select one monster to destroy, I have 2 monster, 1 unaffected by card effects while the other can, can I select the unaffected?
Yes you can.
You can select it and when the effect resolves it will not be destroyed, as The Ultimate Creature of Destruction made it unaffected bu such effect.
And would the same apply if I activate a card that makes me destroy one monster I own and I select the monster unaffected by effects?
If you monster is unaffected then the effect will resolve and such monster remain unaffected same as explained before...
However, let's imagine the card targets and the monster can't be targeted, then that would be something you can't do. However, this is not the case for The Ultimate Creature of Destruction, which "only" makes your monster unaffected by card effects (contrary to, say, making him un-targeteable).