I can’t say what Brynhildr’s current population is, where it stands compared to other servers, or what the most populated server in FFXIV is now. In the past, the server with the highest population was usually Balmung. I haven’t been keeping up with LuckyBancho’s unofficial censes he does lately. Every so often, he’ll release unofficial surveys listing speculated populations and active character counts for each of the servers. They’re usually posted here or on FFXIV’s subreddit.

The servers of Balmung and Mateus are the RP-heavy servers. Both are on Crystal. You can world visit them from Brynhildr. Currently, both are Congested, so you cannot transfer a character or make a new one on them at this time if you are interested in moving to them.

I’m not sure which data center is the most PvP heavy. Specific server doesn’t matter as much since PvP participation is Data Center-wide (along with everything else one can do via the Duty Finder).

East coast versus west coast also doesn’t matter. All NA servers are located in California.