This is quite a bit of a long shot.

(I presume you do not have any similar mouse troubles in X windows by itself. Because once I set the the mouse to the wrong setting, ImPS/2, instead of the correct one, PS/2, and in X had the mouse keep getting pulled to the lower left part of the screen)

You may want to try to setting some of the environment variables for SDL. Specifically-

SDL_MOUSE_RELATIVE
If set to 0, do not use mouse relative mode in X11. The default is
to use it if the mouse is hidden and input is grabbed.

SDL_MOUSEDEV_IMPS2
If set, SDL will not try to auto-detect the IMPS/2 protocol of
a PS/2 mouse but use it right away. For the fbcon and ps2gs drivers.

(I presume you know how to set environment variables to a setting by either "export SOME_SETTING_OR_ANOTHER=setting" or "setenv SOME_SETTING_OR_ANOTHER setting"? I would presume Fedora uses export, but I'm not certain)

Probably won't work, but worth a try.

I'm interested in getting older dos programs tp work under Linux. It might help if you gave us some detail about how you set it up in Linux.
I'm running Vector Linux 5.SOHO, and right now I have Dosemu set up and running on it, but it only gives me text mode programs. Mostly that's what I need, but it would be nice to run some GUI programs. 😀 DavidM http://HisFeet.net

Go to the downloads and try to find dosbox in a format your distro can handle. Or alternately try to find if someone has a build for your distro available somewhere else.
If not get the source and open it
tar -xzvf dosbox-(version).tar.gz
where you want to make it.

And compile as usual
./configure
make
make install
(I usually omit the last step as I have several versions of dosbox and just run it from where I make it)

Read the README about mounting, etc.
Usually the default is good enough but if not try creating a dosbox.conf to customize by using the CONFIG command as specified in the README.
There are frontends to customize and setup dosbox, but I've never used one so I'm not in a position to say much.
Also at the moment dosbox tends to be more aimed at getting games running, so networking, printing, and communications are unfinished if not missing.

(About dosemu, if your version has xdosemu enabled you may want to try using it for graphics)