Pets should still all be there, but you probably have to log in once before you can see them again, they are account wide, and the list fills with every char you log in on.
leveling is still as it always was, fast untill you get to the latest expansion, then the amount of exp needed skyrockets (but that will end with next patch)
Most night 15-20 players are online, sometimes more, sometimes less, focus is a bit on raiding and challenge runs (very hard mode, timed heroics) nowadays, but I have no doubt you'll get some people for LFD if you ask.
Group quests are mostly gone, but if you need help someone will come to the rescue
Scenarios are small 3-man dungeons that tell you part of the story, they are designed to not need a tank or healer, so there is almost no wait time for them in LFD, but do give some valour points, money and a bag that might contain a piece of gear (mostly blues, but sometimes an epic one)
That email was 99.99% sure a fake, I get loads of them.
You can play the game till lvl 85 without getting the MoP expansion, can even play a Pandaren, but will need to buy it to level past 85, go to pandaria or play a monk
I have no idea how much you'll have to download in total, but my wow-folder is now 24 gb.
But with the smart downloader you can log in and start playing after downloading a few gb of core gamefiles, it will then download in the background based on what is needed.