Tchaikovsky's Violin Concerto in D major OP.35.

Prelude in e minor, Chopin

Schubert, Serenade

Somewhat eerie
L'arrivée à l'école, Les Choristes OST

Super eerie
Higurashi No Naku Koro Ni OST - Main Theme
And everything else from this show. Actually, the other pieces are even creepier

Sad
When the Love Falls, Yiruma

25-1-14-14, Pierre Lapointe

The Chairman's Waltz, Memoirs of Geisha OST

Super sad
Schindler's List Soundtrack - Main Theme

I know, I know, no lyrics, but... we can't even understand them anyway
Mozart, Lacrymosa

I have to break one of your limitations as I think one of the more interesting dream style pieces has some faster tempos. Hector Berlioz wrote his Symphony Fantastique as a way to tell a story with music. The story itself goes through many different scenes, but my favorite is the March to the Scaffold. The other movements should be on Youtube as well, and you can read the program he put out with the music here. For me his music is dreamy/surrealistic (I found and fell in love with Liszt’s Totendanz because of my introduction to this piece). There is quite a bit that goes on in the music, as he represents his love with a piece of the music that changes throughout the entire symphony, and the end of this movement represents his head rolling.

I have to agree with Razornoir about Beethoven’s piano sonata #14 (Moonlight Sonata), as listening to that piece, always sends shivers down my spine.

From game music, I always found Ceremony from Secret of Mana to be quite dream-like.

So many good pieces that have dreaminess, wistfulness, yearning, or sadness have moments where they pick up faster beats or heavier ones (as the music progresses), but they also carry emotions throughout the rest of the piece that fit what you are asking for.

Couldn’t find anything else in my collection that fits the criteria without going beyond what you asked for, sorry =(

Have a listen to the OST of There Will Be Blood. Here are the ones I like most:

Prospectors Quartet
http://www.youtube.com/watch?v=o6cTYWoiLcc
Prospectors Arrive
http://www.youtube.com/watch?v=GccggEtpbtc
Oil
http://www.youtube.com/watch?v=SaZ0r_NQYMM

Some nice, slow electronic music:
The Mirror Formula (DL for free)
http://xerxes-music.com/album/the-mirror-formula

what about something like
mono: http://www.youtube.com/watch?v=89zzHCfZ_zs

or maybe jakob (they do have occasional "heavy beat" though): http://www.youtube.com/watch?v=3gZDm8oKmAg

or red sparrowes: http://www.youtube.com/watch?v=D4s6Oo5T86c