If you want to go into PE, I would start early than late where I've seen some BA's leave after 18 months or straight after MBA; 3-4 years is definitely on the longer side where "more" experience in conslting is not necessarily seen as a plus. The one concern I have is your comment about not having choice on PE projects. If so, that also suggests to me that you live in a geography where PE is not prevalent and that also might limit your access into PE as an exit option.
1. Promotion to Associate is only valuable in the sense that you have validation of your performance (3 out 25 in your BA class to be promoted after 18 months)
2. McKinsey BA is definitely eligible but PE is super competitive and so it would also depend on your broader background. Deal experience is crucial especially if you are applying straight to PE. I get the impression they are more open minded when you are coming out of HBS/GSB
3. I would apply before, during and after MBA where it's all about luck and timing where there are usually very few spots compared to a large pool of strong MBB/buldge bracket bankers/MBA grads
4. I'm assuming you are referring specifically to PE (not FInance broadly). If so, familiarity with basic financial knowledge around the 3-statements and financing deals is critical, building an LBO model and then your DD ability in terms of looking at a business in context of it's external and internal drivers is crucial too where you will come across interviews where you will need to argue for or against acquiring a specific asset
Most importantly, I would first focus on enjoying and thriving at McKinsey then start planning for your exit options., Good luck!