With a cm of play, you've made a good call to stop riding it.
Rebuilding a BB is one of the more potentially difficult tasks in routine bike maintenance.
You'll need to get the removal tools for the BB cups, and yes, the drive (right) side fixed cup is a left hand thread. It may also be quite stuck, needing penetrating oil, a heat gun, and/or a mallet. You'll need to get the installation and adjustment tool(s) for whatever BB you end up buying or salvaging. Most lockrings can be loosened with a universal spanner with one tang. There are several different adjustable cone configurations, though. Some can be moved with a large crescent wrench. Some need a special pin spanner.
Many cartridge BBs come with a spacer and you can use it with 68 and 73 mm shells. But the length could be a challenge. Many old spindles are asymmetric, and you may have troubles getting the chain line and shifting right. You'll need to line it up in person to be sure. This table may help:
https://www.sheldonbrown.com/bbsize.html
Is there a bike coop in your area? They're typically very good at stocking salvaged parts and they can be real cheap. And they can help you with tools and advice.