Edge does use two delays with different settings on some songs. You can hear this the most on live versions of some songs. For example, the outro of Until the End of the World on the Elevation tour. Edge uses two delays (hence his use of 4 TC 2290's and 3 SDD 3000's in his last set-up) to get that big sound that you hear. He has also been know to use different delay times for certain live versions of ISHFWILF.
Tap tempo is a way to "tap" in certain delay times based on the bpm of the song. Some delay units do it using MIDI to determine bpm and then figure out the delay time for you. Some stomp boxes use a manual "tap" method, requiring you to tap in the delay time with your foot as you "tap" to the the tempo of the song (this is done by hearing the tempo and tapping in what sounds right).