Can someone help me make sense of the wiring for these switches?
First off -- using a 3 way to sub for a single pole is probably Code, but not the greatest idea

Electrically speaking, there's nothing wrong with using a 3-way (SPDT) switch to substitute for a standard (SPST) switch provided it's wired correctly (or else it simply won't work at all).  However, it can be quite confusing to folks who are expecting the switch to have ON/OFF labeling.

You are free to use your three-way switches as a substitute for single pole switches.  However, personally, I would play it safe here and get some matching single pole switches to go with your three-way switches to avoid confusion.  (Simply take one of the Leviton three-ways you were gifted to the local electrical supply house and ask them for the matching single pole switch.  Good spec-grade switches are most definitely worth the money, and are only a few bucks more than builder-grade trash that's the cheapest thing on the big-box shelf!)

The copper bundles should be ground, but whoever wired these switches fails at grounding

You'll need to use bare (or green, although bare pigtails are easier to get in some ways) pigtails from the green ground screws on your new switches to the bundle of bare wires in the back in order to ground them as per 404.9(B) of the NEC, as whoever wired the original switches forgot to put them in, and the plastic box that was used clearly doesn't do anything to provide your switches with a ground-fault current path.

Pigtails beat backstabs

Finally, it's better to pigtail multiple wires together to a single wire that goes to a terminal instead of using the backstab terminals, especially on cheap builder-grade switches as builder-grade backstabs are pretty bad at hanging onto wires.  This is doubly necessary in your case, as it appears that the wiring in your box is 12AWG, which the backstabs on your new switches won't accept.

Basically, you want to take the black wire that loops around its terminal on the right switch and goes to the left and wire-nut it with a short length of black wire as well as the other black wire that went to the terminal on the left switch that connected to that wire, and connect the other end of the short length of black wire to the corresponding terminal screw on the new left switch.  Repeat with the two wires that go to the other terminal on the left switch, another length of black wire, and another nut, then connect the other pigtail to the corresponding terminal screw on the new switch.

As to why that one wire was sent to two switches? It's the incoming hot feed for both the left and right switches in the box.

Finally, if you need a supply of wire for pigtails, some THHN/THWN dual rated wire will always work.  (Random wire might not meet Code by way of not being a type of wire found in NEC 310.104.)