Players play in the order that they joined the game and the starting player is chosen at random.
Better functionality:
Once all players have joined the game, shuffle them and let the first player start the game. This has two benefits: the play order is randomized and it is easy to see when the next round starts.
I like it. It is a bit confusing atm. How would it work for doubles and triples games though, people want to play with there partner. Would the order of the teams be scrambled as well as the teams themselves?
I have never played a double or triples game. Your post says people want to play with there partner. By "with", I'm assuming you mean "immediately before or after". By looking at a doubles game, it appears that they currently always do. If that's the case, then it should remain the same. Instead of shuffling the player order, you would shuffle the team order and leave the player order within each team alone -- that may be a strategic descision by that team. So the order that they join the game should continue to be the order within thier team. If I have misinterpreted the way things currently work in team games, then forget everything else I said.
It's not that it doesn't work in freestyle games, it just doesn't apply to them since they have no concept of play order.
I see no reason to change the colors of the players.
I agree that knowing when the next round starts is of little importance. The main point is to randomize the play order (in games that have a play order).
I don't really think your idea is much different for the one we have now, and besides, as has already been pointed out, teams can go together which is a bonus. When I play with 4/6 people we make it so that teams don't go after each other, but are instead spread around. Your way could make the game unfair by randomly selecting two players (for doubles) to go together and the other two teams scrambled up. The way it is now is probably better