Overhand/underhand lying extensions

Firstly do the different grips have any different effect on working the triceps?

My right elbow locks out in a bad position at the top of the extension when I use a pronated overhand grip.

Is it alright to use an underhand grip when doing lying tricep extensions with a barbell? because it doesn't hurt my elbow at all...

What about lying hammer extensions with dumbells or a tricep bar?

Yes, you can use different grips. It will stress the muscles differently, but that is not a bad thing. Try decline, with your underhand grip. Those work well also.