1. That's almost like asking "How did you discover sliced bread"? The music of The Beatles has always been pretty much everywhere, for as long as I can remember. It hard to not hear their music pop up somewhere every once in a while.
2. Either 'Let It Be' because of the atmosphere of the song, it almost sounds spiritual, or 'Here Comes the Sun', because few songs have captured the feeling of Spring better than that song.
3. Absolutely. I think that in the late 60s there was an explosion of good, adventurous new ideas and inspired songwriting. There's something exciting about almost all of the pop music that was made around that time. But by now, people have been creating pop music for about 60 years and almost everything has already been done. And if it hadn't been done yet, it's probably just because it's not a very good idea. The progression is pretty much gone and so is much of the excitement and inspiration. In many ways, music from the 1960s sounds fresher to me than today's music.
4. None in particular. I think that they just wrote lyrics that fitted the music. If you listen to 'I Want To Hold Your Hand' or 'She Loves You', those couldn't have been anything else than love songs. Songs like 'Strawberry Fields Forever' and 'I Am the Walrus' on the other hand wouldn't have sounded right without vague, psychedelic lyrics. It all fits.
5. There are a lot of bands from the sixties (or even fifties) that are far from "largely forgotten". Ever heard of The Doors, The Beach Boys, Jimi Hendrix, The Rolling Stones, The Who, Elvis, Pink Floyd, Led Zeppelin, Neil Young, Elton John, Johnny Cash, Stevie Wonder, Frank Sinatra, Fleetwood Mac, Simon & Garfunkel, Bob Dylan...? But the reason why The Beatles are the most popular band from the sixties now, is probably because they actually were the most popular band in the sixties.