That should be possible.. But remember, that you will need as many MAU/CCU as you got users in your online game later. As long as the game stays small that should be fine and only cost you about 9 to 40$ a month (100 CCU - 500 CCU).
I can't tell you whether using the same Server as the Game server (for small Multiplayer Online Game - not really mmo^^) or using a third party provider like Photon Cloud is better for you. That's something you'll need to check out yourself, according to estimated numbers of players, resources and so on.
Well.. Yes, not everyone uses Team Speak and clearly Clans/Guilds won't need to install it to play the game. See it as a special feature so that players don't need to alt+tab. Of course that could also be done with different voip Softwares, but Teamspeak is (at least in europe) clearly the most used voip Software.