Is it possible to see the other players races once I am in the game in Starcraft 2?
Once you are in game you can click the icon in the top right hand corner labelled "Toggle Alliance menu" to see what race everyone is, including the opponents. If they had chosen random at the start of the game they will show up in game as random. For those opponents you will have to scout to find out what they are.
In the case of seeing an opponents most played race while in game, this is not possible.
You can check the alliances tab to see other players' races when you are in the match.