If you look at the chords of C major:
Cmaj Dmin Emin Fmaj G7 Amin Bdim
Then the song really is in the key of C. The Dsus4 doesn't use any extra notes outside the key of C either (just subs a "g" note for the "f" note in Dmin). The first part of the progression doesn't begin or end on Cmaj, but it is still derived from that key.
The first part "sounds" right (in my opinion at least) because of the root movements from chord to chord. If a there was a bass line played that was just the root notes of the chords, it would be F-D-D-E. The last chord Emin really leads the ear back to the Fmaj chord and makes the progression sound cool and "right" because of the sequence of notes guiding your ear back to the Fmaj chord. The bass note "E" at least somewhat "leads" the ear into the F chord.
Also the Dsus4 chord is basically a Dmin chord with the 3rd (f) dropped and the 4th (g) added, which also really leads the ear into the Emin chord because g (the added note) is the 3rd of Emin, so it makes the somewhat un-traditional progression (in the key of C) make sonic sense on yet another level by anticipating the next chord.
The Edim chord sounds wrong because it has a Bb in it, which doesn't exist in the key that the rest of the chords are derived from. On uke most people play a dim7 shape as a diminished chord, which has yet another foreign note (Db) which adds to the "wrongness" of the sound.
I agree that the first part does have a particularly minor sound to it, mainly because it is 50% minor chords, 25% major chords, and 25% neutral (not major or minor) chords. Its kind of cool in the sense that it doesn't have a really strong final resolution point until the 2nd half of the song.
Anyway, I guess that's my 2cents worth of silliness, and sorry for this analytical overkill. I love getting analytical on music, but I guess at the end of the day its just sound that we are attempting to organize in ways that we like, and there are as many opinions about what sounds good or right as there are people (at least people who may be interested in such things)! I do think its a cool progression though.