Okay I'm incredibly confused myself, and have probably made a mess of the issue, but I'll try to explain it all.

Originally I installed Morrowind + it's expansions, and played for a long time. I could only run under 'administrator' to stop recieving an error. This was not the crux of the problem, but an issue nonetheless.

I, on UESPWiki, stumbled across an island below Seyda Neen on the map. I Wiki'd it, and found it to be a plugin. Interested, at it and the other 7, attempted to download them all from the TES site. That worked fine. I had no clue how to get them to work in-game, so I Googled it, and came across this site: http://www.tweaktown.com/articles/316/the_elder_scrolls_iii_morrowind_tweak_guide/index2.html

Under the plugin bit, I did everything as instructed.
So, now when I go to play the game, if I was to run normally (non-administrative), the plugins would appear under 'Data Files', but because of the error of not being able to run unless I was under administrative rights, that didn't work.
When I ran the game under administrator, the plugins would not appear under 'Data Files'.

The plugins were useless to me, needless to say. But I wanted them. So I looked for help on the error. I came across a few sites before finding the issue here, and I fixed it (by copying the Bloodmoon files to my C/: Morrowind folder).

Using the guide, I did the same with Tribunal (copying the files) but I didn't make new folders for the files, and whilst copying them in I simply clicked 'Move and Copy' (for X many files).

So when I click Morrowind and play (non-administrator), the game works fine, and doesn't register Bloodmoon or Tribunal as even existing.
If I run as administrator it certainly registers Bloodmoon, and gives me a couple hundred errors all relating to Tribunal. Seemingly my error in not making a separate folder for them...

I uninstalled, deleted all my save files (not manually in the folder, just when it said 'Do you want to delete all your saved games?') in hopes it'd wipe everything. I reinstalled everything, being Morrowind, Tribunal, Bloodmoon, Bloodmoon Patch and all 8 plugins.
I had hoped my uninstall would delete what I had done, that being my error in the Tribunal copying. It had not.

Again, like before, running under non-administrative rights gave me vanilla Morrowind. I even had save files, but old ones I didn't want (pre-Bloodmoon/Tribunal). Under administrator, the game seems to work fine. I no longer have save files (deleted upon uninstallation), but I started a new game to test the expansions + plugins. I know Bloodmoon works, the island is there. Tribunal works, I slept until a Dark Brotherhood member assaulted me. [As a note I presume they work, as in the whole main questline, the areas, miscellaneous quests, etc. Possibly naive to think so?] No plugins, despite under 'Data Files' I have 6/8 showing.

Basically, I feel like the problem isn't really about Tribunal as such (correct me if I am wrong), but the plugins still won't work.

Edit: Forgot the question(s)...

1) Will Tribunal be fine? I really don't like to mess around with files in fear of deleting or altering something.
2) Any ideas how to add the plugins to my game under administrator?

Side-note: I double-clicked the bar at the top of the TES Construction Set (File, Edit, etc.) and it disappeared. How do I get it back?
Side-note2: I'm bad with computers, and technical terms, locations of things confuse me, and what not.

It would be helpful to know what OS you're using, but I'm going to guess it's either Vista or Windows7.

You likely can avoid all these problems by checking the http://www.gamesas.com/index.php?/topic/781497-vistawindows-7-problems/ See the "Install Location" section especially.
Turning off your User Account Control (UAC) is also a good way to avoid all the problems with plug-ins not working.
Vista is very silly about administrator things, and turning off UAC really is the easiest way to bypass that stuff (and yes it is safe, despite what your computer will tell you) You can always turn it back on when you're done playing.

Also, when you un-install the game, be sure to completely remove it from your computer. (delete all registries of it, empty your recycle bin)

Hopefully I understood your problem correctly, and this is helpful. Anymore questions, ask again.

Ah yes, sorry. Indeed I am using Vista.

I turned the UAC off, restarted as instructed.

I tested the plugins, it appears none work except adamantium armour. I could spawn it. But not the likes of Domina/Gold armour. Wasn't able to get quests in regards to Tohan's Helm, Master Index. No island with Firemoth Fort on, no 'entertain' option in the Eight Plates and still quiet in the Bitter Coast.

Edit: Adamantium is in Tribunal anyway, which is installed, thus I could spawn. I cannot buy it in the shops though, so the plugin did not work either.
I double-clicked the bar at the top of the TES Construction Set (File, Edit, etc.) and it disappeared. How do I get it back?

How should I go about adding those in?

Well I have to ask, are the mods Activated in your Data Files?
Open your Morrowind Launcher, click on Data Files, make sure the mods have an "X" beside them. (double click them if they don't)


But the rest of the CS is visible?
The window must still be there, just minimized or hidden behind the other windows. With the CS open, Right Click on the "TES Construction Set" at the bottom of your monitor and select "Maximize."

I can't believe it was that simple. For both issues.

Many thanks! You've made me a happy man.

Edit: Tinkering done, errors solved.