As my old 17-inch 4:3-size LCD monitor is literally on the blink now after 13 years (flashing screen), I have to get a new one, and it's very likely that the only ones (readily) available are widescreen monitors. That is 16:9-format screens of 21.5", perhaps 24".
A 21.5" 16:9 monitor is actually about half an inch shorter in height than a 17" 1280x1024 monitor.
So you may want to go with a 24" monitor if you're stuck with 16:9. You might not think half an inch is much, but it always makes me feel claustrophobic to have monitors so much shorter than they are wide. "Short screen" monitors bother me more than I would have thought.
What will happen if I try to play these games in full-screen mode? Will they get stretched horizontally? What if no windowed mode is available? And do widescreen monitors have a way to not only adjust the resolution, but also the screen size to 4:3 to overcome potential mismatches?
You should be able to find a monitor that has controls for "aspect" or "scaling" which will scale the image to fit on the screen. If you have to, download pdf manuals for different monitors and do a search for the words "aspect" or "scaling" (with an English pdf) to see if the monitors have scaling controls. There's always the option to stretch the image to fill the screen, but there should also be the option to scale the image on most monitors. Just make sure you get a monitor with the scaling option or you'll be stuck with a stretched screen or a postage stamp.
Which also makes me wonder about current screen resolutions. If they are at least 1920 x 1080 pixels, while my current 4:3 one has 1280 x 1024 at most
1280 x 1024 is 5:4, not 4:3.
Divide pixel width by height to get the ratio.
1280/1024 = 1.25 (or 5/4 or 5:4)
1600 x 1200 is 4:3 (1600/1200 = 4/3)
The difference between 5:4 and 4:3 is not as obvious as between 4:3 and widescreen so a lot of people don't notice the difference.
will I be saddled with an ultra-tiny screen if the game was designed for let's say 800 x 600 pixels?
It should be possible to get a monitor that will do scaling properly and maintain aspect ratio while increasing the size of the image from 800 by 600 pixels on a 1920 by 1080 screen (postage stamp) to 1440 by 1080 (full height with black bars on the sides).
Is it not possible to find another 1280x1024 monitor?
If you enjoy older "full screen" games, a full screen monitor is better.
If that's not possible, see if you can find 1920x1200 (16:10).
It's less "squinty" than 1920x1080 and less wasted screen space with older "full screen" games.