My door frame is at least 7 inches thick, will this fit it?

No. Mine is 7 and it did not fit. I had to send it back. If anyone knows a product that works, please post link. I have been looking. My door frame is 7" and also has a 5" trim height. That is another factor as there are bars our there that will fit a 7" but only accommodate a 3.5" trim height.

I'm 6 1, will my feet touch the ground? How tall is it exactly

I'm 6'4" and my feet touch the ground. It depends though on how tall your door frame is. For reference, arm length is typically around 1/2 your height so unless you have a 9' door frame, there's not much you can do. All pull up bars will be the same. This is the best one I've used to date though since it gives me a couple extra inches above the frame, so I highly recommend it. just think of it as getting an extra hamstring workout while bending at the knees.

Is this good for underhand chinups also?

yes it is. The only minor issue is the supports are at the spacing I would like my hands so I end up a bit wider or a bit narrower than my ideal. It's actually pretty versatile though offering a number of different positions, including side handed on the support bars that also have grip area.