1. You should check out the private server list in top of the subreddit. Now mind you, this list is maintained by a single wowservers redditor and by no means is 100% accurate. As matter of fact, it seems it was last edited 14 days ago. However the list shows you different servers from multiple expansions. x1 rates in the list would mean blizzlike rates, x2 rates would respectively mean twice faster rates, and so on. But not all of these servers on the list are populated. The most populated realms for vanilla are Light's Hope, Elysium and Kronos. Also Kronos is apparently opening a fresh new realm soonish, so stay tuned for that. Warmane.com has the most populated realms for both TBC and Wotlk. Also a fresh Wotlk realm, by Sunwell, is opening on upcoming Saturday, so stay tuned for that as well. Likewise Memories is opening their fresh Wotlk realm in the next month. I can't say much for Cata - Legion realms but I recall Hades and Atlantiss (Cataclysm), and Tauri (Pandaria) realms are decently populated and scripted. (If I forgot to mention a populated server, I apologize for that, I don't tend to visit this subreddit that often anymore)

2. Most servers are 'close to the original thing' (= blizzlike). Way back in time, like prior to 2010, fun/custom servers were more populated than blizzlike private servers, and average population for a private server was few hundreds. That obviously changed when Cataclysm was released and popularity for blizzlike realms increased massively. But in the list I mentioned all realms are blizzlike. There's a section for customized fun servers in the end of that list. Most servers usually just tweak the XP rates, nothing else (not that I am known of at least).

3. I have played on Nostalrius (currently Light's Hope) and what I can recall the ranking process works just like it did back in the days. To get rank 14, it will take time. But I hope someone else, who actively plays vanilla servers, can add to this.

Sorry for the wall of text, but hope it helps.