The resolution of games on a real DOS machine is just controlled by the games themselves - at the time CRT monitors were the norm and they can display a variety of resolutions without the ugly scaling you can get on an LCD, so games were developed to run on any reasonably standard resolution and nobody minded.
When you use an LCD monitor instead you have to rely on the electronics in the monitor to do something useful with the signal. Some monitors are better than others and may offer controls to tweak the aspect ratio, while others may just stretch whatever resolution you give it to fit the screen. Stretching not only means that the image can be the wrong shape, but if your monitor isn't an integer multiple of the original resolution then each pixel's colour gets smushed across more than one real monitor pixel, and everything looks awful. Some LCD monitors will do this better than others, it all depends on how good the electronics in the monitor is. For these reasons a lot of us keep a real CRT monitor for running older systems.
If you must use an LCD but don't like the way it looks you can get better results using DosBox on modern Windows. Graphics cards now can do a really good job of scaling low resolution screen modes to the native resolution of your monitor, and keep the aspect ratio right.
In any case, you're not going to cause any damage to your PC or monitor - worst case is that some resolutions may not display and you get stuck on a blank screen. There's really not anything you can do on the PC to make it output as 4:3, because it IS outputting 4:3. To make it display in 4:3 on your 16:10 monitor the computer would have to output a 16:10 image with the 4:3 image in the middle - this is the graphics card scaling I mentioned, but you'll only get it in modern Windows.
