Are there any other men's IIHF games besides the different divisions at the Worlds? Are there international IIHF single game friendlies like they have in soccer?
Yes, quite a few actually.
I'm not sure if you're just asking about teams from lower divisions, so I'm gonna mention even those played by elite nations.
For starters you have the tournaments like Euro Hockey Tour (Sweden, Russia, Finland and Czechs) and Euro Hockey Challenge (a series of friendly games played between top 12 Euro team from IIHF ranking in the weeks leading up to the Worlds), Switzerland, Slovakia, Belarus, Norway etc. also play small tournaments during IIHF breaks such as Arosa Challenge or Slovakia Cup.
Then there's the Euro Ice Hocke Challene which is a series of tournaments, which features some of the lower ranked teams from elite level and Division I teams:
http://www.eihc.eu/view/eihc
Most of the games are played during the official IIHF breaks and leading up to the Worlds.
There are some excpetions as well though, for example France sends a u25 (or might be u23 team) that plays pre-season games with Slovak league teams or Poland has also sent a NT to pre-season club tournaments.
But I don't think there's a lot of the friendlies bellow the Division IB level, if for no other reason because we're talking about countries where hockey is a fringe sport and the federations don't have enough funds for these kinds of events.
EDIT: Also as has been mentioned there is teh Challenge Cup of Asia for the Asian teams that don't play the Worlds, but that's more like their version of the Worlds than friendly games.
And if I remember correctly there was an international tourney in South America featuring NTs last summer, which included some team from Canada as well.
And of course, this system differs a lot from the one FIFA has as the differences between the level of play in hockey and in soccer differ hugely. You may see a football team of let's say Luxembourg play against one of the better football teams and while the score won't be all that pretty, it won't be that bad either. If let's say Finnish NT played the Luxembourg hockey NT we would talk about them losin by dozens of goals and I'm no even trying to exaggerate (and that woud be Finland playing with their Euro selection, while in the FIFA friendlies it's hardy unusual to see the best players playing, we'll hardly see that in hockey due to the NHL vs. international hockey rift).