Questions about moving from MBB to MM and/or LMM PE

I'm in the process of exiting Bain (I'll refer to it as MBB from now on so that others searching can see the keywords) have a few questions about moving from MBB to MM/LMM that I haven't been able to find clear answers to, and I was hoping you monkeys could help me. Don't feel obligated to answer everything if you don't have the information on each piece--semi answers are helpful here as well.

1a) How difficult is it to get into HSB/GSB from an MBB and MM or LMM background, and if it's harder, why? I know that MBB has ~3x the chances of getting into B School at any given MBA program (Nearly 100% for my MBB), so if it is harder, why do the odds go down?

1b) Is there a big difference between LMM and MM when going to HSB/GSB? If so, what is the magnitude of the difference, and how do you overcome it?

1c) If you do not go to HBS/GSB, how difficult is it to go from the other M7 into a Senior Associate/VP Level in a PE fund?

2a) How often do you travel in LMM or MM funds? Is there a difference between the two?

2b) When you do travel, what is it for? Mainly board meetings, discussing new strategies, etc? (Assume that this is pre-COVID)

3) Is the interview process different if you are MBB? I've heard some say that they're lighter on technicals and modeling, but others say that they're actually much more stringent--what has been your experience?

4) In LMM or MM, how often are you able to work side-by-side with the management team and improve the business, rather than just monitor? 

5) Last, which firms are overall best for an ex-consultant that wants to focus both the modeling/investing side of business, as well as the operational/margin improvement? No need to be exhaustive here, but would love to hear a few examples.

Thanks!