OK, so here's the thing:
Recently, I began downloading Sims furniture, hair, makeup, and whatnot from the ModTheSims2 site and whenever I try to add all of my downloads to my game, it winds up glitching and terminating the whole game.
Is there a way that I can add the downloads to the site where the furniture and stuff is by deleting the mandatory Maxis furniture and other Maxis stuff like makeup, hair, etc.
Can I do this without damaging my Sims 2 cd altogether?

Lemme get this straight. Are you trying to add the downloads to your CD? Because that isn't how you do it.

In your "My Documents" folder, there should be a folder named "EA Games". In that folder you will find a folder called "The Sims 2". In that folder, if you haven't already done it or it isn't there yet, create a folder called downloads. That's where all your downloads should go. Have you been doing that? If not, that's how you do it.

For the love of all that is good and pure, DO NOT DELETE ANY OF THINGS IN YOUR GAME! Keep it as is. If you get default replacement stuff, it will override it for you properly.

Sorry if I misunderstood your question. I hope I helped!

Please read the FAQs over in Game Help on MTS. There are full instructions on how to install custom content the proper way.