Welcome to the forums! Some guesses:
I forgot what L2-FIX.COM fixes (over the vanilla L2.EXE). It might well have been a saving bug. Are you running L2-FIX.COM (and not the vanilla L2.EXE)?
When you try to save in L2's save menu, you must enter a nonempty name for the save slot, then hit enter.
The Lemmings 2 directory should not be write-protected in the host system. Assuming you're on Windows: In Windows explorer, right-click the directory -> properties, then disable write protection for the directory and for all files in it.
Contemporary Windows systems don't necessarily allow write access everywhere. Instead of having C:\dosgames, have the Dos games somewhere under your user directory (websearch for where that is depending on what kind of Windows you have). Move the entire dosgames directory into your user directory, then tell Dosbox that new location to mount. (E.g., if you followed Kieran's tutorial and added mount instructions to autoexec, edit the mount line in Dosbox's autoexec.)
-- Simon