Your initial instructor certificate is always done with the FSDO, unless the FSDO is too busy to accommodate your needs. At that point, the FSDO will assign a DPE to do your check ride. Unless the school you attend has signing authority, which is pretty rare (and expensive).

CFI Guy is correct. The majority of your CFI work is self study. I'm sure you have CFI friends who can give you guidance during this self study phase. I would not even entertain the idea of jumping into the airplane until you have thoroughly completed your ground study/preparations.

You can accomplish the CFI relatively cheap if you're smart about it. If you decide to jump into these puppy mills to get your CFI, well, pay up. I think it would benefit you to talk with a CFI buddy. Or ask questions on the forums. At the end of the day, what you need to know is in the CFI PTS, which is substantial in your knowledge base.

Getting the MEI first is an option. But I think it's a really poor option. Say you pass your MEI. Guess what? Just about EVERYONE starts in a single. That makes you a MEI without many students. Not too many people are getting their multi these days now that the whole airline career is all but a waste of time to most people to chase after. So the action is at the single-engine level.