I'm not sure how Impulse works, however. I bought the retail version of the game, apparently it doesn;t come with any copy protection when on disk and doesn't need impluse.
Whether you buy the retail box or download the game through Impulse, patching is still handled through Impulse. So you'll need it to update.
Can I register that with Impulse, then pay for Entrenchment, then download it?
Yes.
Can I backup Entrenchment once that's done? Or is Impulse just a store thingy that I can buy a download off?
By right-clicking any installed game on Impulse, you can choose to archive it. Usually a good thing to do before updating to a new patch in case something breaks or you still have old save games you want to finish.
Is it just an executable I download, or is it updated into the game?
Impulse will help you install Sins 1.15 and Entrenchment separately so you can play either. They will show up as two separate games in your start menu and Impulse.
Does it just work like Steam where I register my copy?
Pretty close. Nothing radically different.
Finally, can I play it offline?
Impulse doesn't ever have to be running to play Sins or Entrenchment. It's primarily for updating, creating archives, and purchasing other titles.