Does my 2012 Cylinder #1 look OK?

IMHO; at that oil consumption rate; I'd be looking for a fully rebuilt engine with factory recommended modifications/updates. At this point, a head gasket job is ONLY a band-aid to a much larger issue. That oil burn is slowly choking-off your CAT. If you haven't been flushing out, replacing your brake fluid every 5 years, the ABS pump and valve assembly would be vulnerable - another known expensive repair for gen3's.
It may be time to start looking for another car.

Sorry...

 

With a failed head gasket and near runaway oil consumption, my shopping list would be:

Daihatsu 11400-37140 - BLOCK ASSY, SHORT
Daihatsu 90910-02166 - BOLT (FOR CYLINDER HEAD SET)
Toyota 04111-37316 - GASKET KIT, ENGINE OVERHAUL

 

I hear where your coming from. I do wrench on cars and there are a lot of older, reliable, gas only cars that are cheap and easy to fix. That old mechanic friend of yours may be able to give you some pointers. I'm partial to Toyota's and Honda's; but got a really good deal on an old GMC Envoy 4x4 that gets 17 mpg. It's pretty much a tank and drinks gas like one - but it was free and great for hunting, fishing, and camping. Just a set of coils and passenger side wheel bearing @ 250K miles. Toyota n Honda will be in the low to mid 30 mpg. I've crunched the numbers since I live in one of the most expensive gas states here. Even for me the hybrid break-even point is past 5-6 years, so you shouldn't rule out a straight gasser. The newer the car, the better the mpgs. The only reason I'm in a Prime is because of the major discounts and incentives given during COVID - otherwise I'd be in a gas-only Corolla or CIVIC. That's what I was looking for, when someone totaled-out my daily driver. All those discounts and incentives brought the car's price down to those gas-only; so the extra mpgs was just a bonus for me.

Hope this helps....

 

Knowing exactly where you are helps when asking for "who" to use advice.

1. Does it make sense to invest in a replacement head gasket...

In my situation, no. But your situation is different. Personally I like to have a newer reliable car if I keep an old one.

2. Who do we trust - from whom to acquire a 'rebuilt' or 'updated' engine - or to have this one rebuilt, and what's the going price?

JDM used in Japan engines from a JDM engine distributor, ideally picked up and inspected rather than shipped. A rebuilt shipped from a reliable source like Hybrid Pit in Southern California. But I would not do either unless I had a hybrid shop to lead the decision making at your age and skills.

3. Here in SE-Georgia, USA - who to trust to remove the old engine, and then to install/reinstall the rebuilt/replacement, and adjust everything, correctly?

The hybrid system really does not need "adjusting" with an engine replacement. Its actually less technically challenging than a head gasket job where valve train and timing is removed and has to be reinstalled accurately. Replacement is within the scope of most shops as long as they don't dump coolant in the ac compressor's connectors and are not irrationally afraid of the orange cables.


This is where I always recommend finding a smaller hybrid shop even if it means driving an hour or more.

In many situations, finding another car makes sense especially if your car has not replaced the hybrid battery or brake booster yet. The oil consumption problem won't go away either and could take out the catalytic converter or even seize the engine unexpectedly if the 4 quarts total were used up. Which could happen within a 5k mile oil change interval.

Hopefully someone here knows a good shop in Georgia or North Florida. I hear about Tampa Hybrids (4 hours from Waycross) and see videos from Royalty Auto Service in Kingsland Ga (probably dealer pricing but perhaps good advice if local to you). Or have a dealer quote something (but it will be a new short block and rebuilt head in most cases). I have seen them suggest a US salvage used engine in situations with financial stress. That should be a hard no.