Converse style flat shoes are a good bet. No heel drop and a moderate amount of ankle support. Probably not so good for running. I wear Nike essential 3s to the gym, and they serve me well enough, though they're a bit prone to slipping if I go in from the rain. The best non-slip shoe I've had is the Reebok Work n Cushion, but they've got a bit of a drop. Very comfortable shoe though, probably would suffice for all your machine and lifting needs.
I've heard good things as well about vivobarefoot. No drop at all.