Some more input/feedback...
1) Are omnis appropriate for hat mounting at all, regarding PAS technique?
To expound on what others have said, I suggest that you read all you can about how two mics are used to create stereo effect... and understand the concepts and then apply them to your recording situations. For example, learn how directional mics (cardioids, subcards, hypercards) can create a stereo image on a recording by 'hearing' differences in sound pressure levels between on-axis and off-axis sounds resulting from placing the mics in various configurations (ORTF, DIN, DINa, etc), while any pair of mics (directional or omni) can create stereo imaging by recording differences in time by having the mics separated on the horizontal plane from each other.
Hat mounting is very appropriate for omni's because the omni's end up acting similar to your ears when it comes to 'hearing' what is being recorded. Since your ears are located on either side of your head, about six inches apart from each other, facing directionally opposite from each other, one omni microphone on either side of your head does essentially the same thing. The stereo image is created by two things...1) your head separates the mics by six inches (17cm) so that sound coming from your right gets to the right mic a fraction of a second before it gets to the left mic and vice versa for sounds coming from the left getting to the right mic. 2) your head is also a baffle so that sounds from the right of the room gets somewhat attenuated in the left mic, and vice versa. Both of these things cause the recording to recreate the imaging you experience in the live setting by creating tiny differences in sound dynamics between the two channels, so that you'll hear the stereo imaging when you listen back later.
2) Let's say I'm in a club that's standing room only. If it's not especially crowded or busy, do you just place your mounts or tripods in the center of the room, pointing at stage? I'm concerned about it being annoying or a tripping hazard to other attendees. With a Tascam, a tripod, and the band/venue approval, what other places could I put this recorder that would make sense, including the stage?
YES! As a taper, you MUST get comfortable putting your stand up in the middle of a venue if you want to get the best sound possible...or as others have suggested directly in front of the board. I mean, it's just something you have to get used to. It comes with practice, but you learn to grow a pair of balls and kind of walk in like you own the place...ESPECIALLY in situations where you know either the band or the club are taper friendly. The best thing to do is to get there at doors and very first thing you do when the doors open is set up your stand. That way you're already staked out your territory, not many people are there and as the crowd develops, they just naturally form around you. If the audience is there first and THEN you arrive, sometimes people will give you the stink-eye because they don't want a stand in their sight lines...but those same people are usually the ones that ask you for a copy of the recording later, so once again, just get there early and set up. After you do this a few times, it gets easy and you don't think twice about it.
3) I recently picked up a mannequin to try out HRTF-style recording at DIY spots. Are omnis or cards more appropriate to stick to its "ears"? I'm aware that bringing in a janky mannequin is not going to fly with too many venues without some very sweet talking 
Instead of a complete manequin, you can buy just the styrofoam head and use those. The band moe. always uses a styrofoam head to record the ambient sound during their shows and then mixes that into their SBD recordings that they sell on-line. The head has a name, but I don't remember his name. He has his own pair of sunglasses and over the years, he's grown a mustache. LOL. He got kidnapped at one point, but I think I read that he got returned...or maybe the kidnapped one just got murdered and moe. ended up replacing the original one.
4) What would be a decent upgrade path for my gear? From lurking, I can see that the Zoom H series is not very popular due to its noise floor. For now, I'm more concerned with being able to ditch the hat mount and am looking towards things like this mount so I can have a beer without worrying about self noise 
Once I went from stealth recording to stand recording, my enjoyment of shows increased exponentially. I do still stealth shows, but it's SUCH a pain in the ass compared to shows that I can record open. I target bands that are open taping so that I can enjoy it alot more. But yes, get yourself a stand and mounting gear, put your mics up on the stand and enjoy the show.
The other thing about stand mounting is that you can manage crowd noise. By raising the mics about 3 feet or so above the heads of everyone, you effectively put chompers into the background, especially if the PA volume is pretty loud.
The only negative about stand mounting is that you have to pay attention to people tripping over during the show. It depends on the specific venue and show, but if there's a band that I really want the best sound possible, I'll set up dead center and deal with blocking all night. I'll take my flashlight and courteously light people's way as they walk past my stand so that they can see and don't trip over the legs. If the band isn't that important or if the sound is really great in a venue, I'll set up back by the board and generally don't have to worry about the stand as much if it's out of the way. Some FOH guys will let me put my stand inside the cage area if there's room. That's really nice because in that case obviously I don't have to worry about trip hazard at all.
And yes, being able to be free to grab a beer or three makes all the difference in the world to your enjoyment of a show. Yeah, I want to get a good recording, but I'm there to enjoy a live show as well.
5) What resources are there for mixing AUD/SBD sources? What process do you usually go through when mixing AUD/SBD sources, or multiple AUD mics? My mixes so far (in Adobe Audition) have been just lining the sources up as best I can, maybe some EQing/envelope tooling, and just putting out what sounds okay to me (through headphones and cheap nearfields). I've received advice to split my stereo AUD tracks into mono, hard pan, and then EQ according to the instruments that are emphasized in each individual mono track - but it's a lot of work, and my attempts to do so usually end in a worse result. How meticulous is your process? Do you outsource mixing to other people?
If you have Adobe Audition, you have everything you need. If I record by the board and get a board feed, I'll use one of my multi-track recorders to record both mics and SBD onto the same recorder. That way all tracks are time synched, though I might need to slide the SBD track back a couple of milli-seconds so that it matched up with the ambient tracks (since the tracks have latency between them). But if your sources come from two different recorders, yeah you'll probably have to use Audition to stretch or shrink one of the recordings to match the other. It's easy to do in Audition...just use the stretch tool and zoom in to find a drum beat at the beginning of the show (and synch them up in Audition) and another at the end of the show and align them up using the stretch tool. I find that you might have to do the stretch a few times in order to get perfect alignment, but it doesn't take long. You also might need to break both recordings down by tracks and then align them track by track. Sometimes I've had drop-outs on one or the other recordings so in that case, a track by track alignment is probably needed,
As far as your friends suggestion to EQ according to instruments on each mono track...fuck that! Too much work, unless you happened to have recorded, say, eight tracks with all of the instruments on stage being individually mic'ed up and recorded onto their own ISO track. But if you just have an AUD and a SBD, to me the biggest decision is to how much to mix either recording in the final mix. Like for example, should the final have 40% SBD and 60% AUD, 50/50?, 70/30?. Audition is easy because you can preview the mix before rendering the final master so that you decide what sounds best.
And just so you know...you're DEFINITELY not alone when you mix your recordings down today, then come back tomorrow and completely hate what you ended up with today. It's so funny how that works. You'll spend two hours with a damn recording getting the EXACTLY PERFECT mixdown between AUD and SBD, and then a day or two later you come back and listen and say WTF was I thinking? LOL...
Good luck man! I'm jealous that you're just at the beginning of a VERY fun journey.