Great question! Let's take a look:
You may send one of your dudes at the deed home. If Diamond is at the deed, he can choose himself, satisfying the effect on the second sentence.
Move Diamond to another location. Diamond will move to another location regardless of the choice of dude for the second sentence.
Correct, it says 'you may' send a dude home.
The 'deed' for all effects refers to the deed Diamond was at when the ability was first activated, for clarity.