I have copied the plugin files I downloaded from the official site into my data files folder as instructed. However, I don't think they have actually "plugged-in". I have the Game of the year edition. Do you have to install the plugins prior to starting a game? For example, their is no adamantium armour at meldor's in Balmora (adamantium plugin), and there is no "entertain the patrons" option at the Eight Plates after installing the "Entertainer" plugin.

Help!

Yes. Start the game and click "data files" in the opening menu. You will get a list of your plugins. Double click the plugins you want to play with. You should see a checkmarks at them on the list. Quit from the data files and start the game. The plugins will be loaded. (Note: the same way you "uninstall" a plugin. Double click the plugin so that the checkmark vanishes, etc.)
Usually, you will get warning messages about "master files", which do not correspond to the plugins. You can usually ignore them and click "ok". You can get rid of them by means of loading the plugins (along with the Morrowind master file) into the game editor and save them.
Also, a new plugin often needs starting a new game. Otherwise you can encounter problems during the game. You will see. Try continuing the game and if you run into serious problems than cannot be solved by means of a console then start a new game.

Is there any chance the Game of the Year edition has been streamlined to allow the insertion of the plugins more easily? (i.e. after you have started a game) Will removing the plugins overcome any problems, or will my game be ruined if problems do occur? (I don't feel like starting over again!)

Uninstalling a plugin during a game can result in problems, especially, if the saved game depends on the plugin (plugin-specific items in the inventory or in your chests, and so on). It can, that means, it need not.
At any rate, save the game before you start adding or removing plugins. In the worst case, you'll be always able to return to your saved game (after restoring the original situation as far as the plugins are concerned) if problems occur.