Is it possible for an individual to start their own blockchain working with Bitcoin Core? Is it possible for an individual to create their own blockchain from 
Bitcoin Core?

I think the value of BTC is currently controlled by miners in China and I want to create a blockchain with either a filter on who can mine or a limitation on the amount each individual can mine.
It is indeed possible, with some tweaks in the code to differentiate from the original. Effectively, you'd be starting an altcoin with all the same parameters as bitcoin.

However, if it had everything exactly the same as the original, it would be very vulnerable as some big bitcoin miner could one day wake up and decide to point his hashpower to this new coin and perform a 51% attack.

Note that this new x-coin would be worthless unless you could bring enough people to actually use it and enough miners to secure it. But if it's a copy-cat of bitcoin, why would anybody come?