* Does it always have a socket? - Yes, it always comes with a socket, even at level 5. This is very unusual because usually helmets get sockets approx. 10-12 levels later (there are some exceptions but nothing that early).
* So, should I just start running Campaign to get the crown as early as possible? - Absolutely most definitely NOT. The helmet is useless without a gem, and when leveling you usually don't get a ruby before level 30 (you might get lucky and find it earlier, or not until you're 40; I've done several test runs and have never found a ruby before 30). However, the lower the crown the bigger the stats loss - which can be quite significant, if you're wearing a "useless" legendary for 25 levels. If you want to get the crown, get it when you get a ruby. You can run to Leoric and "set up the waypoint" or have a friend do that on season start, but do not make the mistake to run around the first hour of S7 with a terribly underperforming item with a useless legendary effect.
@Bagstone, where can i find this FAQ you referred to?
Also a little tip: gems start dropping at level 19. Easiest and surest way of getting gems is doing a bounty boss. The diabolic chest always drops a few gems, so a couple of these and you will surely have a ruby.
I just made this up, this was my little FAQ, containing 2 questions that I see a lot, hence "frequently asked" ;-)
If I start in campaign and kills enemies in a natural progression, I usually get the crown between 25-35 minutes. I am normally lvl 12-14 when this happens.
Will have to do some testing to see, but I have a feeling that it is faster just doing adventure mode from the get go yet given the low quality of gem that you will be throwing into the helm.
Also, in 2.4.1 testing I was able to hit lvl 70 in 2hrs and roughly 30 minutes solo using no legendary items. By the time 30 minutes hit, I was usually around lvl 18. Compared to lvl 12-14 when going for the helm, would be interesting to see how the stat decrease of the helm but addition of a multiplier to the gem affects the process.