Regarding the etymology, I read a number of years ago that "Tattoo" (which in British usage is a military display or performance - cf. the Edinburgh tattoo) comes from the Dutch "Tap toe", i.e. close the taps on the canteen beer barrels. It would seem reasonable to assume that the American name "taps" refers to the same function.

Regarding bugles and trumpets, in the British Army the bugle was used for signalling in the infantry, whereas trumpets were used in the cavalry (both are still in use for ceremonial purposes by the appropriate regiments). I believe that drums also had a signalling function, though I don't know if this predated or complemented the use of wind instruments.

Regarding use in funeral or memorial services, the practice in Ireland (and I imagine at least some other countries of a predominantly Christian tradition) is to play the last post followed by reveille, representing death and resurrection.