Just installed the Morrowind Code Patch, and when I tried to save, it said I had reached the maximum number of saves. When I open my save folder (C:/Program files(x86)/Steam/Steamapps/Common/Morrowind/Saves), I see a list of my saves, but the ones titled "Nirillor(save#)" (as in, "Nirillor0001", "Nirillor0002" etc.) only go as high as "Nirillor0049 even though the in-game save menu says it goes up to "Nirillor 78". (Nirillor is the name of my character, in case you were wondering.)

Also (and this is the big problem), I have deleted about two dozen saves, but the in-game save menu still says I've reached the maximum, and several of the saves I deleted still appear in the in-game menu. What is going on here? How do I fix this?

Probably because you have MW installed in Program Files.
You can either move your install, or disable your UAC (User Account Control)

This step is needed for anyone using Vista, Win7 or 8.
It allows you to have control over the files, and ensure all mods will work correctly.