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The Grim Resolve detachment for Dark Angels got an additional rule that states:

In your Command phase, select one Adeptus
Astartes unit from your army; until the start of your
next Command phase, add 1 to the Objective Control
characteristic of models in that unit.

If I put that on one of the planes, can I potentially grab objectives with them?

And if I use the plane in hover mode (making it lose the aircraft keyword) and give it +1 OC, can I do actions with it?

I don't see why not.

It is a non-Aircraft model with an OC that is not 0. Assuming it isn't Battle Shocked, it may make an action that does not have other limitations it does not meet.