ish,
you have opened up a HUGE can of worms here!
it's really hard to explain this type of stuff succinctly, but i will try!
song ideas can come from ANYWHERE. a song can start in your head, with a guitar riff, a drum beat, bass line, sample, from jamming on another song, or a number of other places.
for example, the edge came to the joshua tree sessions with a demo of "where the streets have no name" the basic guitar intro was there and so was one other chunk of the song. however, there was a REALLY odd timing drum part that Edge did with a drum machine. the band voted that out and worked on the rest of the song together with brian eno & daniel lanois. they spent about half the recording time of The Joshua Tree writing and recording that one song!
"Pride," on the other hand, developed instantaneously during a soundcheck jam before a concert. Bono has said that the actual length of that song was the amount of time it took to write. "One" was another tune that developed super-quickly, although it happened in the studio during the Achtung sessions.
as far as mixing goes, that's a WHOLE other process. when a song is recorded, a band can record upwards of 50 tracks of various instruments, sounds, and vocals for it.
mixing is the art of bringing everything together into a rich stereo picture that helps convey the emotion(s) the song deals with. many parts are used sparingly or altoegether left out. whoever mixes the album can make or break it, especially with a band as great as U2, because they are so dynamic lyrically, musically, and emotionally.
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"I still haven't found what I'm looking for."