TLDR: Reliance transfer switches don't play well with GFCI/AFCI
I'm interested to know the power requirements of the furnace to see if I can also run from a 200W battery pack. My plan was to set the transfer switch to Generator, and then plug an extension cord from a house outlet, through a Kill-a-watt energy meter, and then into the transfer switch, essentially using my house wiring to stand in for the generator.
That seems like a reasonable approach for testing. Here's the problem.
And it's 1 problem. You don't need 3 questions here.
These Reliance style transfer switches, including this one and the 6/8/10 circuit transfer switches, have a very hinky and 1970s mentality way of handling neutral. They simply treat neutral as if "it's all going to the same place, so it doesn't matter". But it does. With GFCI it definitely does.
The hot wire is doing exactly what you expect. it comes out of a house circuit, through Kill-a-Watt and extension cord, to transfer switch inlet, thorough the switch, back through the panel and into the furnace. Fine.
In a 21st century sensibility, the neutral wire should follow the exact same path in reverse. Thus, the two opposite-direction currents on that path are equal, magnetic fields are not emitted, and GFCIs do not trip.
However, if this was installed according to instructions, the furnace neutral simply comes from the panel neutral bar, end of topic. And so the big loop of neutral, from branch circuit to Kill-a-Watt to extension to xfer switch to furnace, is not happening at all.
The GFCI sees dissimilar current (hot current out, very little neutral current returning) and assumes the unaccounted-for current is shocking a human, and trips the circuit. Well done GFCI.
TLDR: if you want to test That 70's Transfer Switch, you'll need to use That 70's Circuit to supply it. Lose the GFCI.
The problem is, this "alternate" routing of neutral may poison the results from the Kill-a-Watt. I don't know how they work internally, if they measure neutral, they won't see the neutral side current and that would throw it off.
Issues with the installation
Now as to your installation, the cardinal rule of neutrals is that neutrals MUST go on the NEUTRAL bar. Neutral is not ground. Neutral handles regular service current and must be thermally rated for it. Ground only handles fault current, so liberties can be taken such as ad-hoc improvised ground bars like in photo. (I have one in my own panel so that's fine if mounted correctly).
In a MAIN panel, and I presume yours is the main from the labeled main breaker at top right, grounds are permitted as guests on the neutral bar since the neutral-ground equipotential bond is in this enclosure.
Some who have worked too long in the industry and forgotten their schooling think neutrals and grounds are good on any bar. No, neutrals on neutral bar only. Or possibly your guy was not a licensed electrician.
Code vio #1: neutral on an accessory ground bar. Also a 110.3 violation since it was not installed per instructions.
Code vio #2: that particular ground bar/dual lug is not rated for >1 wire per lug (unless it is, but I doubt that). That was field-installed but not by your guy: it was done by someone previously, and they did 1 wire per lug.
Root causes
The root cause for the mischief is that your neutral bars appear to be full, which isn't possible because UL doesn't approve panels that don't have enough neutral spots for every circuit to be 1-pole - and you have five 2-pole breakers taking 10 spaces but using at most 4 neutrals. So you should have loads. However, read your panel labeling - many panels allow 2 or even 3 grounds to share 1 lug on the neutral bar. UL assumes you are exploiting this fully.
So option 1, exploit this fully.
Option 2, install accessory ground bars - consult the panel labeling again and it should list certain model numbers of accessory ground bar which are designed to fit pre-drilled/tapped sites on your panel. Get the biggest ones. Ground slots are cheap. There should be several sites, mount 1 high and 1 low.
And then you can de-commission that field-applied ground lug down below, or simply empty it out, since it can take larger wires than most ground bars can. Make sure it's attached with a proper fine-thread screw, and not a sheet metal screw.
Other cleanup considerations
By the way, since your main breaker is a "backfeed style" breaker mounted in a normal breaker position, have you thought about getting a generator interlock for this entire panel, instead of mucking about with problematic Reliance gear that mishandles neutral and won't play well with GFCI/AFCI? This style is fairly easy to get a generator interlock for, since you just mount the generator breaker across from or next to the main, and a sliding plate interlocks the two. Siemens' version is under $30 and even straps down the breakers (though it does not fit this panel). I would talk to a GE dealer about such an interlock for your panel.
The electrician probably didn't mention this because these factory interlocks are cheap and there's more markup and commission on Reliance.