Is this normalization correct? (two many-to-manys connected by a many-to-one)

Question

I have a schema for a scorekeeping database with Game, Team, Player tables.

One team has many players, each player has only on

Answer 1

The problem to your design is that you have used surrogate identifier as the primary key for the tables, a well defined primary key will solve the problem:

Team   -> pk:team_id
Player -> pk:player_id
TeamPlayer -> pk:{team_id + player_id}

Game   -> pk:game_id
GameTeam -> pk:{game_id + team_id}
GamePlayer -> pk:{game_id + GameTeam_pk + TeamPlayer_pk} 
              = {game_id + {game_id + team_id} + {team_id + player_id} }

Having check constraints on GamePlayer will help the problem:

GamePlayer 
{
 Check game_id (FK of Game) = game_id (FK of GameTeam );
 Check team_id (FK of GameTeam) = team_id (FK of TeamPlayer);
}

So
player_score will be property of GamePlayer.
team_score will (may) be SUM of GamePlayer.player_score with specific team_id.