The 93 crankset might play nice with an HG70 chain (it's borderline, but usually ok), but many 7-speed chains are narrower at only 7.1mm width. With the stock 52X45t chainrings the Shimano HG chain is usually ok.

Find a 7-speed Shimano cassette road wheel and remove a 1mm washer from both of the axle's spacing stack to maintain wheel dish.
Measure the inside dimension at the dropouts, it's usually 121mm at least. The narrowed (124mm) axle will slide in easily w/o undue stresses.

You may need to trim away at the driveside dropout adjusterwith a file or Dremel to clear the chain and lockring. No biggie there.

You will need to add a notch under the derailer hanger tab for the B-tension screw to puch against, and tap threads into the hanger's mounting hole.

You can use a 13-34t cassete with a mtb derailer for wider-range gearing.

RSX shifters are somewhat precious on the market, expect to pay $100 plus for a pair of 7-sp STI shifters. You would need a downtube cable stop to go with those.