yet they are complex and mysterious and rich
Um, no, not the first two. They're entirely appropriate to the song, which is the key part. That's not about chord progressions though - it's much more about fingerpicking and strumming patterns. This may lead to some effect of inversions if you look at the theory, but that only derives from standard picking patterns.
Changing the tuning can give you different inversions for chords, which perhaps gives a sound you're less familiar with. Dylan learnt the altered tunings he used from Joni Mitchell, a much better guitarist and more sophisticated melody writer, and didn't tailor his own tunings to the songs in the way that Mitchell did. If you want to hear the effect of altered tunings, definitely look into her work.
Drone notes can give you some interesting textures too, and they're often a feature of altered tunings. If you're not used to the effect of drone notes, perhaps that sounds mysterious. It's a very standard part of folk music though, to the extent that the distinctive sound of Scottish bagpipes comes from their drone pipes.
The songs may sound rich with good production too. Don't underestimate the effect of double-tracked guitars, which is a standard recording trick. If you've recorded the same guitar in the same space with the same settings, and the multiple parts aren't hard-panned to left and right, it's nearly impossible to tell which take is which - you end up with something which sounds like a single hugely complex guitar part, when actually it's just two (or more) much simpler parts. For an obvious example, Lyndsey Buckingham's fingerpicked guitar on Never going back again can only be approximated live, including by Lyndsey Buckingham (who's said he had to rethink his arrangement to actually perform it), because it's actually at least 2 guitar tracks overlaid.
Dylan's chord progressions though, they really don't venture far at all from the standard I-IV-V (G-C-D) formula and their relative minors (Em-Am-Bm). Sometimes he adds circle-of-fifths moves (G-C-F), and sometimes he uses a second which is a very standard feature of blues (G-A, and especially G-A7 where the G remains in the seventh chord), but anything outside of that is very rare. Nothing in this is anything a beginner guitarist shouldn't already have encountered in their first 6 months of learning the instrument. Perhaps you'll see them first when you play a Dylan song, but you can go back to the earliest blues recordings and hear all these, or go back 300 years and find the same moves from JS Bach and Mozart's studies for beginners. Dylan was particularly influenced by Woody Guthrie, of course, like all folk musicians of his era.
Long story short, if you want to work out how Dylan wrote his guitar arrangements, start learning guitar and you'll know as much as Dylan within 2-3 years. It really isn't much more advanced than that.
Now if you want to replicate Dylan's lyrics, that's a whole different can of worms. Dylan's guitar work might not have been radical, but his songwriting was well out there. The chords for Don't think twice, it's alright could have come from anyone, but the lyrics are genius. They look simple, and they are, but it's the kind of simplicity that comes from being exactly right.