Should women bother working out their chest?Working out your chest won't decrease your breast mass (don't I wish, tho). If anything it might increase it slightly, due to more prominent pectoral muscles underlying the breast tissue. Losing weight will make your breasts shrink a bit; losing a lot of weight will make them shrink a lot. Some people lose weight when they start lifting, but not everyone does, and it's not necessarily everyone's goal. Some people gain weight.
Working your chest will help you support yourself, a little, in a handstand, but it won't make you a better balancer, and there's a whole bunch of other muscles in play for that move. The closest maneuver to simulate a handstand is a dumbell military press, which hits the pectorals only as a secondary muscle group. It's especially important to work out your forward "pushing" muscles if you're planning on working your backward, "pulling" muscles. For one thing, separating your workouts between mostly push and mostly pull will allow you to work out more frequently without exhaustion. In other words do pullups on Tuesday and bench on Thursday.
I couldn't imagine a comprehensive workout that didn't involve the "big lifts" for the upper body, including some kind of bench work for the chest and triceps. If you don't have a compelling reason not to work out your chest, I'd recommend you dispense with this silly notion. Your strength and fitness gains will be a lot better if you don't wimp out by concentrating solely on the ancillary "beauty" muscles.