@SephReed: I'm not sure how more modes would sound in general, but one of the ways you do get more modes is by having more strings. Because the strings will not be perfectly in tune with each other, you will get "beating" as they go in and out of phase with each other. I would describe this as sounding thicker or richer. For a piano, the low notes usually have one string, the middle notes two strings, and the high notes have three. You can hear the (sudden) differences as you go up or down the piano, although the piano will also switch between wound and unwound strings at some point.

It is not strictly true that the decay will be exponential if the oscillation is small enough to be linear, because the envelope will contain all the different vibrational modes, and each mode has its own decay constant. You can see this if you plot the envelope on a log scale, and the effect is not limited just to near the attack.