We initially used phpBB, but switched to SMF because SMF supported addons and phpBB did not. This was a long time ago.
We use oneall.com for the social login, which integrates with phpBB as well.
All forums these days support a store feature of some kind. SMF integrates directly with Paypal as a first party feature now, I believe.
We have no custom user groups/permissions these days, though we used to integrate the SMF groups into our Garry's Mod server.
We use dreamhost.com for our hosting. We've had some issues with them, but not as many issues as we've had with other hosts. No host is perfect, and you generally get what you pay for -- do not look for the bottom dollar. Most hosts will also sell you a domain, we get ours from dreamhost and have enjoyed their price and service. Just don't buy a domain from godaddy (horrible company).
If we were using a CMS, I guess we could easily have the forums at ulyssesmod.net directly. Mostly a style choice I suppose, though most communities with a forum that I know of use forums.xyz.tld.