You do not need an MBA to be a PM. Unless you expect other serious ROI on it (which is plausible), I would save yourself the time and money.
I moved from EM to a PM role. Did this at a startup where I expressed interest/aptitude in product, and they let me transition over. I recently moved on to another startup where I was hired as a PM.
It basically came down to showing interest and effort, taking on product type tasks even without the title or permission, and forming relationships with decision makers.
Answer to #3 highly depends on what you want your days to look like and what you want to achieve. Both roles require you to talk to a lot of people, cross-functionally. PM you'll likely have more say in the business strategy, direction, and more interaction with other disciplines (e.g. design is just as important as eng to a PM). EM's will require more people-management, like 1:1s, mentorship. Expect to be your team's leader, therapist, and professional unblocker, even if that means running to the office supply store to get them another whiteboard.
If I could reduce it to its simplest terms, as a PM you will have more impact on the product (duh), and as an EM you will have more impact on the people on your team.
Both seem equally likely to lead to VP roles - to me that will come more down to your reputation and relationships than your specific job title.