I've never had much success with getting mods to work on games. I tried to get three mods working on Morrowind, but I ran into two problems.

The mods in question are code patch, MGE XE, and Morrowind Rebirth.

I thought I had things squared away, but I went to play through... and the game never started. I figure maybe I deleted something while I was adding files, so I verified. After that, every time I would click Data Files (in order to see if my mods were there), the launcher would crash.

Now, I have uninstalled and cleared out the mods from the folder. I want to go through and try it again, so can anyone give me some pointers? Did I do the right thing when copying the items in the mod Data Files into the main Data Files folder?

How did you install the Code Patch? It has to be executed because it is not a mod. Does the game work with the Code Patch only? Does the game run with the Code Patch and Rebirth but without MGE XE? What do you mean by that? Was it a crash to desktop? If so, when did it occur? Did the screen remain black or did nothing happen at all? You should reinstall your game and keep one backup of the new install. That sounds right to me. You also have to activate all ESPs and ESMs in an order where every ESP gets loaded after the ESPs that it depends on. Tribunal and Bloodmoon have to be activated, for example.

I added the Code Patch contents to Morrowind folder and then executed the program. I know the game played at that point. I went back and did the same with MGE XE before I tried to add Rebirth.

I was having success which is why I got careless. It never did anything. The wheel span for a while like it was trying to execute while it stayed on the desktop. The crashes that came after verify were from launcher to desktop with a "Check for Solution" prompt from Windows. This might be my problem, because I don't fully understand what you are saying. My expansion were active, and I did get the process of activating my mods so maybe my order was off.

After executing the program, did you actually tell it to patch your game? Did you disable some of the patches?

If the screen never went black after starting the game, I think the problem is caused by something else than the ESMs and ESPs (the files that you activate in the launcher).
My best bet would be that something's wrong with MGE XE, but you say that it ran with MGE XE before?

If I were you, I would reinstall the game just to be safe and test it with Morrowind Rebirth and the Code Patch but without MGE XE.

Assuming the game does work then, I'd install MGE XE and run the game directly using the (new) Morrowind.exe file (not the launcher) as administrator.

It would be a good idea if you posted your load order here, that is the name of all the mods activated in your launcher in the same order they are listed there.
Without your load order it is difficult to tell what might be wrong with Morrowind Rebirth.

If you installed the game in your Program Files directory, that can lead to problems but GOG's installers default to a different directory, anyway.

You can also see and alter your load order in your Morrowind.ini file. To the bottom of the file there are the names of all plug-ins (ESMs and ESPs) and you can add a new plugin by inserting its exact file name.

I don't know why your launcher crashes but something that can cause the game to crash and all sorts of other problems is that your expansions are listed after the ESM of a mod.
Your official content should in almost all circumstances be loaded before anything else, so everything else should be further to the bottom of the Morrowind.ini file.

Find this in the ini. Should I change the order manually?

[Game Files]
GameFile0=Bloodmoon.esm
GameFile1=Morrowind Rebirth 3.7 - Racial Diversity 1.5 [Addon].esm
GameFile2=Morrowind.esm
GameFile3=Tribunal.esm
GameFile4=adamantiumarmor.esp
GameFile5=bcsounds.esp
GameFile6=master_index.esp
GameFile7=Morrowind Code Patch Showcase.esp
GameFile8=Morrowind Rebirth 3.7 - Balmoras Underworld 1.0 [Addon].ESP
GameFile9=Morrowind Rebirth 3.7.ESP
GameFile10=Siege at Firemoth.esp
GameFile11=Morrowind Rebirth 3.7 - Morrowind Patch 1.6.5 Beta [Rebirth Edit].esm

Absolutely. There are some things extremely wrong with it. Maybe your launcher crashes because of your last file. Your game shouldn't run at all or even allow such a load order, so, I'm not sure what's up with that.

Always read the installation instructions of your mods as a beginner, I can see that you didn't because the Showcase file is active.

I'm not sure if I fixed it, but I think I corrected the worst errors:

GameFile0=Morrowind.esm
GameFile1=Tribunal.esm
GameFile2=Bloodmoon.esm
GameFile3=Morrowind Rebirth 3.7 - Morrowind Patch 1.6.5 Beta [Rebirth Edit].esm
GameFile4=Morrowind Rebirth 3.7 - Racial Diversity 1.5 [Addon].esm
GameFile5=adamantiumarmor.esp
GameFile6=bcsounds.esp
GameFile7=master_index.esp
GameFile8=Morrowind Rebirth 3.7.ESP
GameFile9=Morrowind Rebirth 3.7 - Balmoras Underworld 1.0 [Addon].ESP
GameFile10=Siege at Firemoth.esp

It works. It crashed once and was slow to start, but it seems to be working now. I did notice that some of my Data Files were not active, so I am not sure if that needs to be changed.

http://i66.tinypic.com/oshd1c.jpg

I didn't tamper with it yet because last time I couldn't get back into Data Files after I changed things.

Not to sound defensive, but I couldn't have made it as far as I did without following instructions. The only mention of Showcase in the instructions was that it was optional. I often find the directions in these readmes to be vague, so I appreciate the help all the more.

Do you have any idea on how I got my order mixed up so bad? Is there a certain step (perhaps the load order in MGE XE) that I might have blundered to cause that? For future reference, of course.

Thanks again.

A slow start is completely normal with many mods. If the crashes happen whenever the memory usage is high, there is a patch that enables Morrowind to use more gigabytes of memory. With what? With the load order in the .ini file? I take back what I said. I don't have any good explanation for it. There are a few other things that I don't understand either. For example why the Morrowind Rebirth ESMs (Master Files) are deactivated in your launcher if they were listed in your .ini file.

What do you mean by load order in MGE XE? Do you mean the distant land creation? No, the only difference that would make is that objects would look differently in the distance than they do when you're near them.

A few hints:
Keep in mind that the .ini file can get overwritten, so make a backup of it before and after you've made changes.

I think Gopher from Gophersvids on Youtube has enjoyable instruction videos regarding load order and the basics of modding Bethesda games.