e.g.:
https://www.ft.com/content/b0a6481e-d20a-11e6-9341-7393bb2e1...
http://www.telegraph.co.uk/technology/2017/01/04/mark-zucker...
I don't think that he'd want to manage a company that just fights off rivals. Becoming a politician and (trying to) make a change sounds more like him. He has the money, people mostly like him and he'd only have to find a CEO for Facebook to get rid of conflicts of interest, as he isn't involved in many other businesses.
Facebook could always give away VR headsets for free (disguised similar to the Free Basics program). Facebook has already run out of space to put ads on the Facebook platform, and so it has started trials on Messenger (in a couple of countries). One way to put ads everywhere is to install large screens (like in Sci-Fi movies or like those in Times Square). Another way is to give a small screen to people that's always in front of them and put ads there based on what the person is looking at, the location, time of day and other factors - that's what VR, or rather AR, will be about whenever Facebook gets serious about it (in relation to Wall Street and earnings). Google Glass probably had the same long term intent in mind when it was developed.
He already has a CEO: Sheryl is more than capable of running the business. Sure, she'd need a tech-visionnaire CTO (like a Jony Ivy for her Tim Cook-style), but I think FB would be perfectly fine without Mark.
I think that depends on geography and demographics. He's popular in SV and the tech community, but perhaps not as popular elsewhere. It's hard to find objective data on it. The only thing I could find was a silly cbinsights bracket vote. [1] It would be interesting to see what his Q score is.
Associated hn discussion: https://news.ycombinator.com/item?id=13371991