I'm not sure how it works in the GOTY edition, but if you purchase the expansion packs separately then the missions will appear in your quest log automatically (after a text box pop-up describing some radio signal or something similar).
The only expansion which ties into the main story is Broken Steel - it is an extension of the main storyline beyond the regular ending, and allows you to continue playing after completing the main quest line (unlike the original).
The others are all totally standalone, all taking place in alternate locations to the Capital Wasteland, and so it makes no difference to the main story when you start them.
In terms of toughness - Point Lookout definitely has the hardest enemies to fight, so I would tackle that one later on. The others are fairly playable at any decent level I think.
This problem has different solutions depending on how you want to play.
If you're just looking for the achievement, one solution to your problem is to do whatever missions you want, choosing the side of good, until you get relatively close to your level point. Then go into a town and steal (for minor karmic reduction) or lay waste to entire towns (for major karmic reduction). You can quickly lower your karma this way. Then, once you're neutral, go kill some random creatures until you level.
(If you're looking to steal, a good place to do it is at Dukov's place… he's got stuff everywhere.)
The "nice" thing about this system is that if you stay on the path of "good" the whole time, you can do this at each achievement point: be bad until you get to the right karmic level, level up, get your achievement, and then restore a save to get back to your good self. You can quickly get "good", "neutral", and "evil" achievements for all levels on one playthrough this way.
(I did this, because I don't really enjoy long-term playing as an evil character… but I do like my achievements!)
You could also do this as an evil or neutral character, but it's much easier to quickly drop from "good" down to your new goal, than to work your way up.
If you find that you're close to leveling, and you're at the right karmic level, and don't want to take a chance on changing your Karma by doing a story-type adventure, you can always go wandering looking for random bad people to take out. Going near the Washington Mall can get you some XP fairly quickly, and you can go inside the capitol building and clear out large areas without picking up any Karma, positive or negative.
Best Answer
Either choice you make in the rotunda will "kill" you. It will show you the black and white film and apparently "end" the game even with the expansion. However, after the "end" finishes, it should allow you to continue (unlike without the expansion, where I believe it just takes you back to the main menu).