What are those chords?

I'm used to reading chords the other way:

e <=high e string
b
g
d
a
e <=low e string
and based upon your descriptions of the chords it looks like that's what you intended but are your string labels flipped backward? At any rate, as written these chords are:

1. C major 7 /E
2. Probably an f major 7 shell voicing with third omitted, but with only two notes it depends a lot upon the context in which you're playing the chord. See the note below.
3. Probably a shell voicing f dominant 7.
4. Probably a shell voicing for Eb major 7.
5. Probably a shell voicing for Eb dominant 7.

Now if your string labels were flipped and your high "e" string is marked correctly, here's what we are looking at:
1. D9. (You note correctly that the 5th is missing. I leave out 5ths a lot. They are largely unimportant to chords unless you're playing altered chords or you're going for a certain type of sound.)
2.  Another shell voicing, could be Em7 but depends upon context. Could also be a G major.
3. Could be Cm7 or Eb7.
4. Dm7, again depends upon context. could also be an F chord.
5. This might be a Bbm7 or Db7.

Hope that helps. Sorry to be ambiguous but when you're dealing with simple two-note chords the devil is in the details.