Is running multiple 12/2 home runs through one single hole against code?

The issue you're running into involves NEC Article 310.15(C)(1), which covers adjustment factors for more than three current-carrying conductors in a raceway or cable. When you pull four 12/2 Romex lines through the same series of holes, you have eight current-carrying conductors bundled together. Once you hit that many wires in a single space for more than 24 inches, you are required to apply **derating** factors beacuse the heat can't dissipate as easily as it would with a single cable.

Technically, 12 AWG copper with THHN insulation is rated for 30 amps in the 90°C column. When you have 7 to 9 conductors, you have to derate to 70%, which brings your allowable ampacity down to 21 amps. Since you are likely putting these on 20-amp breakers, you are still technically under the limit, but it's cutting it close. Many inspectors will take one look at four cables stuffed into a 1-inch hole and fail it on the spot under the "neat and workmanlike" requirement of 110.12, especally if they look cinched or twisted.

If you haven't stapled them yet, my advice is to just drill one more hole. You've allready got the drill out, and it's much easier to add a second path now than to rip everythign out after the inspector puts a red tag on the job. Space the holes at least a few inches apart to maintain the structural integrity of those 2x10s. Keep your holes in the center third of the joist depth to stay within code for structural boring.

And don't forget about securing those cables. You need to staple within 12 inches of the panel or any junction box, and then every 4.5 feet along the run. Since you’re going through the joists, the holes count as support, but you still need those staples at the ends. If you just leave them hanging in a big loop where they exit the joists to enter the panel, that’s another easy fail. It’s better to do the extra work now than to have to explain yourself to an AHJ who thinks you're trying to take shortcuts.

I'm curious about that 24-inch rule for the bundling. If the cables are only touching where they pass thru the actual joists, but then they spread out in the empty space between them, does that still count as a continuous bundle? My joists are 16 inches apart, so there's a good bit of air in the bays. Also, how close can I drill the second hole to the first one or to the plumbing? I'm already worried about making the joists too weak with all the holes for the pipes and wires. I'd hate to fix the electrical issue just to have my floor start sagging later.

The 24-inch rule is a bit of a gray area depending on the inspector, but if you end up using fire-stop foam or if those bays get packed with insulation later, that heat won't have anywhere to go. In those cases, they'll definitely count it as a continuous bundle.

For the second hole, keep it at least 2 inches away from the first one and any plumbing lines to keep the joist's structural integrity. I usally stagger mine vertically by an inch or so instead of putting them perfectly side-by-side. It makes pulling the wire easier anyway since they won't be fighting for space at the same angle.