My rear wheel is in the shop so I borrowed a friend's. Problem is, my bike has 8-gear shifters (the kind with thumb triggers and numbers) and an 8-speed 12T-34T cassette, while the borrowed wheel has a 7-speed 11T-24T cassette. It was the best I could do on short notice. Both wheels are 700c / 28 mm.

I tinkered for quite a while with the high and low limit screws and the fine tuner thingamabob. My initial approach was to lock out the "1" gear and adjust to where gears 2 through 8 matched the available gears on the casette. Then I switched to locking out the "8" and matching up with gears 1 through 7. For whatever reason, I couldn't get either of these approaches to work at all. The shifting was horribly erratic.

Finally, I managed to get it set up to where the "1" setting fits nicely into the biggest rear gear, and the "8" setting matches up with the smallest rear gear. The shifting is smooth enough, but the shift from 3 to 2 doesn't do anything, nor does the jump on the way back up from 1 to 2. In essence, gear "2" is MIA in the gap between 1 and 3.

I am heading now to pick up the repaired wheel, so the issue is moot. But I'd like to know whether there is something I could've done differently (other than buying a spare wheel or borrowing a wheel from a different friend!). Might chain length be part of the issue?

Also, I'm curious whether what I did might have caused any damage to the wheel, cassette, derailleur, etc.

You did not mess anything up but it sounds like the drive trains are not compatable. On indexed shifting you have to use the same number of gears. It is possible to get two separate cassettes with different # of cogs but it is only for an emergency will not shift correct. You can solve the problem easily you need at least 2 bikes.

If any, minimal. The goods news though is, you didn't end up with the derailleur in the spokes.

Might I suggest in the future that you consider a second bike - lower cost, simpler...for exactly this reason. A backup.

Most cyclists I know own two bikes - with the second usually a single speed/coaster/old 3-speed/junker. But some do have their second being similar to the primary just because they can.
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The other option is to have a spare rear wheel - but that means insuring both hubs mirror each other so that swaps can be made with minimal derailleur adjustements.

=8-)

I think the reason it didn't work is because the 7 speed cogs are spaced further apart then the 8, so the chain needs to move more left & right with each shift. But the shifters are for 8, so they don't move the chain enough.

People use 7-speed cassettes/freewheels with 8-speed shifters all the time, and the shifting is acceptable for most.
I have even raced on such a setup for many years.

What WASN'T done in this case was to back out the B-tension screw to match the smaller freewheel.

The enormous chain gap that resulted from the smaller cogs made the derailer's motions very ineffective at moving the chain from one sprocket to the next.

One more consideration would be if the 7-speed freewheel was truly index-compatible, but the B-tension screw still needs to be set with the top pulley close to the sprockets in any case.

I've experienced this exact problem. Ran a 12-28 (or was it 11-28?) cassette on a mountain bike once. The cassette profile was flatter in the middle than the arc the mountain-derailleur moved on. Despite backing out the b-screw, I could never quite eliminate the large gap and resulting finickiness in the shifting, especially in downshifting. Some downshifts in the middle of the cassette range required some finesse, some extra holding of the thumb, some minor overshifting. As dddd says, it was all down to excessive chain gap.