Firstly, thanks very much to those of you who guessed, I realize my song list was probably a lot harder to guess than some. It certainly garnered less answers than ladypixel's.
On to the answers!
1. Come, just as you are to worship.
"Come, Now is the Time to Worship" from Brian Doerkson, I think, off of that Time Warner collection of Christian Music you always see advertised on Cartoon Network. Yeah, that one. And hey, I LIKE that song...
2. ...but square cut or pear shape, these rocks don't lose their shape.
"Diamonds are a Girl's Best Friend" by Marilyn Monroe. No one sings it like her.
3. And then one day on the hit parade was a little dark-haired boy who played the Tennessee flat top box.
"Tennessee Flat Top Box" by Johnny Cash.
4. Te wo nobaseba ii, yonde mireba ii, I love you, I love you, soba ni iru yo.
"Daydream/Sobi Ni Iru Yo (I Will Be at Your Side)" from the Kimagure Orange Road soundtrack...
5. Power of love meguriaeru, itsudemo kanjiaeru.
Yet another anime one where I left the title in to help you....still, no one got ANY anime ones....*pout* This is "Power of Love" from Master of Mosquiton
6. Tobenai haadoru wo, Makenai kimochi de kuria shite kita kedo, Sutaato rain ni tatsu tabi ni obiete ita.
More anime, although this time no title, cause I figured the words "Yuzurenai Negai" would be a dead giveaway, but apparently I should have left them. That's the title, by the way, it's the first opening theme to Magic Knights Rayearth.
7. Change all of your grey skies, turn them into gay skies, keep sweeping the cobwebs off the moon.
Yet ANOTHER with the title in it, and still no one got it. I figured because of it's age I had damn well better leave it in....It's "Keep Sweeping the Cobwebs off the Moon" from Ruth Etting.
8. You're so very special, I'll have that special.
I chose this lyric specifically because it's about the only line that differs between the two versions of "Creep." This line would indicate that it's NOT the original and actually from Richard Cheese.
9. ...show you all the beauty you possess, if you'd only let yourself believe...
Adia from Sarah McLachlan. One of the few cases where it was truly my favorite line.
10. Try to remind myself that I was happy here, before I knew that I could get on the plane and fly away, from the road where the cars never stop going through the night, to a life where I can watch sunset and take my time.
Yet another Truly Favorite Line here. "Sand in my Shoes" from Dido.
11. Raion ga unaru mitai na raimei ga, Taisan o unagashiteiru, dou naru no? Sou... mou sugu ame.
Damn, I SERIOUSLY expected robyngoodfellow to get this one, I got the darn song from her. "It's Gonna Rain" from Rurouni Kenshin. Don't you ever sing along to it??
12. If we were made of cellophane we'd all get stinking drunk much faster!
I love this line!! Squirrel Nut Zippers doing "Ghost of Stephen Foster." Surprised no one got this, SNZ is relatively popular and how do you forget a line like that?
13. It helps me forget him, so the louder the better...
Uber-congrats to huai_hsing for getting the song, although the "him" is a giveaway that it's a remake and not the original.... "Turn it on, Turn it up, Turn Me Loose" from Heidi Hauge who has an album which seems to consist mostly of kickass remakes....
14. You can sit around and stare at the picture tube 'til your brain turns into cottage cheese.
NO ONE GOT THIS!! ARGH!! "UHF" from Weird Al. It's from my favorite movie folks!
15. Hold me, love me, hold me, love me, I ain't got nothing but love, babe...
Gotten more than once, and damn right it should be, this is classic Beatles doing "Eight Days a Week."
16. Sayonara lonely in my life, Itsudemo happy day my life.
Not surprised this went unguessed, even with all the English... "Jump/Meippai Dakishimete" from the Hand Maid May opening.
17. I am human and I need to be loved, just like everybody else does.
THIS IS NOT LOVE SPIT LOVE DARNIT!! There WAS an original and people do still listen to it! Sorry, tired of all the remakes making me feel like a fossil. The Smiths doing "How Soon is Now?"
18. You know just what I was there for, you heard me saying a prayer for someone I really could care for.
"Blue Moon" from the Marcels. Another classic. Fell in love with it at the end of American Werewolf.
19. In the lonely light of morning, in the wound that would not heal, it's the bitter taste of losing everything that I have held so dear.
Played every hour on the hour by every Clearchannel affiliate and no one got it. Pity. "Fallen" by Sarah McLachlan.
20. (Couldn't pick one line on this one, so this one's easy...)This pill will help me yet, as will these boys gone through like water/I've spent so much time living in survival mode.
Yet another one I'm surprised people missed, with two lines yet. "Precious Illusions" from Alanis.