California Dreamer by Wolf Parade

Please help me analyze this song. I took a whack at it, but I'm not super confident that my analysis is spot on.

https://www.youtube.com/watch?v=3CAEzB3JzSA https://tabs.ultimate-guitar.com/w/wolf_parade/california_dreamer_crd.htm

So it seems like the song is basically in E Harmonic Minor with some exceptions. Instead of a 7th degree of D# diminished (D#-F#-A), they flat the seventh and use a D Major (D-F#-A) instead. Also, instead of an augmented chord on G for the third scale degree (which WOULD be III+ = G-B-D#), they alternately either sharp the third and have an augmented chord on G#, so III+ = G#-C-E, or they actually just use a major III (G-B-D - the seventh is already sometimes flatted anyway, right?)

scale: E-F#-G-(G#)A-B-C-D-(D#)

*i: E-G-B

*iio: F#-A-C (not actually used in the song)

*III+: G#-C-E (or III = G-B-D)

*iv: A-C-E

*V: B-D#-F#

*VI: C-E-G

*VII: D-F#-A

So the intro is just E minor

The verse is VII-VI-i(x2), VII-VI-V-i(x2)

The pre-chorus (I'll be around) is iv - III+ - i

The chorus (And I think I mighta) is i-VII-VI-III-iv-III-i

How did I do? Is this how you'd analyze it?

And what is your opinion on the "function" of the two different incarnations of the third (III+ and III)? I have very little experience with using augmented chords, I generally avoid them because I feel like I don't understand them. They definitely have an uneasy / unsettled quality to them.

Also, for people with good ears, what is going on with the more melodic elements of the song? Are they generally using G or G#? Are they generally using D or D#? Or a mishmash of all?

Lastly, how would you analyze the chords that are not naturally in the scale? Are they "borrowed" or is this mode mixing or what?

Many thanks for any feedback. I'm trying to get past the phase in my theory knowledge where it's all "theoretical" and based on the book learnin' and out into the real world where I try to understand the songs that I loved before I knew a damn thing about theory.