I usually did back stroke in tri's until I became a life a life gaurd and was really taught how to swim, I could only butterfly and several other strokes for short distances. got punted in the head and shoulders several times, But was fast enough on a bike to make up for it massively.

as for the bike, while you might not be putting weight on the broken hand, you will be losing a brake and a shifter for a while,
and if you live in a flat area, that may not be an issue, but there may be a situation that comes up you wish you had both.
when I burned my thumb, I just let it heal for a bit. before I started again.