Will this bracket fit correctly, and tightly on the 60d with a Bower battery grip (BOWBGC60D)?
Yes, providing you tighten the screw it will fit tightly.
Will this bracket work with third party battery grips such as Vello? I have a 60d with a Vello BG-C6 battert grip on it.
Hi Brian, if you want I can create a video so you can see and get a better idea of what I mean. I can do it this weekend
Will this bracket work on a Canon 50D with grip?
While it's possible that this unit may fit the 50D to a certain, this particular bracket is designed around the 60D and as such, is not recommend for the 50D. The openings for external connections as well as general fit will likely be less than ideal when used on the 50D.