Would a doctor be able to see if you had arthritis or gout from an MRI and xrays?

Yes for arthritis. The x-ray will show joint destruction and bone spurs from arthritis. All types of arthritis will eventually show bone and joint changes on x-ray. Initially gout will not show any bone changes on the x-ray, however, if you have recurrent gout in one or more joints you can develop gouty arthritis which will show up on the film.
Probably. It would depend on what type of arthritis you are asking about, but typically many types of arthritis will be seen either on plain x-rays or mri. Gout is not as likely to be seen on x-rays or MRI with many variables to consider such as is this acute gout, chronic gout, have tophi (large crystal clusters) formed. You might even see inflammation in gout on MRI but be able to tell what it's from.