There’s an outdoor swimming lake in Passy, but Chamonix town also has an outdoor pool. You’re in France mind, so speedos are obligatory. This is not a joke!
Early July is not too busy, August is when there’s no point riding most trails due to the number of other folk, but Holy trail and Merlet are both out. Goat trail (new name on me, but it’s the only trail I can think of from the mid station that get ridden often, heads to the village of Lavancher and runs parallel with the Pas du Chevre so I guess that could be the name origin?) is an official trail from the Grand Montets mid station, so is allowed, but the lift company is replacing one of the chairs in that part of the mountain, so the trail might be closed. Petite balcons sud & nord are both marked trails so you can ride them too.
Good news is that the trails from Le Tour back to Trient & Chatelard (NOT aig du Posettes) and from Les Houches are not included in the Chamonix bike ban, so are all good.