The stick moving back and forth while in gear is the bolt securing the shift lever to the trans is loose. A very simple repair (once the console and boots are removed).
Jumping out of reverse can be caused by many things; shift lever hitting shift boot, engine mounts, trans mount, worn reverse gear fork, etc.
the transmission mount seems to be a very weak bushing which if worn could cause the engine shake. that one can be kinda tuff to change if you don't have some good tools and/or thoughtfull ideas for removal.
i had a problem with my shifter popping out of third because the stick was hitting the center counsel, just needed some minor adjustments.