I haven't seen all the animes you have, but I feel I can still give a few suggestions. And I'm not sure what kind of music you are into, so I'm gonna through what I know at ya...
My Town by Hollywood Undead seems to be a good fit for FLCL. Or ya could make a good mix with this song. Summer Wars and FMA might help with that.
https://www.youtube.com/watch?v=Nf1gBSvl5NgImpossible by Manafest. I hear this song, all I can think is Full Metal Alchemist. It could be an AMV with great development throughout. Give it a look!
https://www.youtube.com/watch?v=u6vAy_G8aYAJars by Chevelle. Read the lyrics and Gundam just pops into my head. I haven't completely seen a whole gundam series, but this is the first thing that comes to mind, or some big Mech anime should work.
https://www.youtube.com/watch?v=DtvfoRq-55gLast Man Standing by Pop Evil. Would make for a great fight AMV. Could do one anime dedication or a mix. Either would work.
https://www.youtube.com/watch?v=jDmGu0y973UThis is Gonna Hurt by Sixx:AM. Another song with a lot of potential. Could do Evangelion. Maybe Negima. Or something with a lot of fighting like one of you mech series. Maybe. IDK, lol
https://www.youtube.com/watch?v=eIhnrYYvA4gWait for Me by Rise Against. There is simply too much you can do to this song. This song it the shit. Listen to it. Love it. Imagine the possibilities. (I'm a big rise against fan, I'm a bit biased.)
The Messenger by Linkin Park. A good song for a softer AMV. Would go quite well with Kanon or most of others animes like it(Air TV, Clannad, H2O etc)
https://www.youtube.com/watch?v=0H-FeeMJ0fMThe Deep End by Crossfade. This could go great with Evangelion. Probably works with a couple others you have too. Like I said, I've only seen about half of what you have listed.
https://www.youtube.com/watch?v=eKzc48Ddu9EI know how ya feel bout not having much time or motivation to edit. Life is complicated >.< lol