Hey Mike - it’s much simpler than you’re making it here my friend.
If you’re playing with a guitar who’s doing a I IV V in key of E (and even if there’s a vi in there) there’s 2 good options for you to explore:
1.) E Harmonica playing in 1st position. BOOM.
2.) A harmonica playing in 2nd position
Now, as you expore both of those over songs like the Green Day song you mentioned, you’ll find certain notes don’t sound good at certain times. To get out of that scenario, you may want to invest time into learning the Major Pentatonic scales which you can just play throughout the whole tune and should sound great;
One more note here: learning the melody will always serve you when soloing. You can play/embellish the melody for a solo and you’ll always be loved for it.
Hope that helps! Rock on! 