Hi there,

First of all, best of luck to you - I really wish the US would make it easier for skilled immigrants to enter, though it sounds like you're on a great pathway already.

1) Do MBB apply for a green card? If yes, at what level do they support this (e.g., senior consultant, engagement manager)?

My understanding is yes they do. In terms of the exact process/conditions I highly recommend you speak to someone at these firms (HR/recruiting and/or LinkedIn networking)

2) Due to the immeasurable benefits of a green card and the added industry experience I build as I wait for the card for ~3-4 years, I should probably transition to consulting only after I obtain it. Some international ex-employees of my firm have transitioned after a few years and my guess is that they had the green card when they did it. However, if get an offer at that time, will I still enter at the associate consultant level? 

Instead of guessing why don't you reach out and talk to them!

You would enter at the level that makes sense based on your level of experience. The level you enter at has nothing to do with whether you need a green card or not! It has to do with years of experience, MBA, historic roles/responsibilities etc.