> how transformers work in general when they're connected backwards.
This is a complex question and I should be in bed.
The (typical) 20% regulation works forward or backward. So 20%*20% is 40% regulation and a very significant drop.
The Turns-Ratio usually allows for regulation. A "120:12" is probably wound for 120:14, with a few ohms resistance, so that under full load it hits its 12V spec. When you work backward, this step-up becomes a step-down.
True, a 120:12 - 12:120 rig should self-correct at no-load, but at full load it will sag bad. Because of resistance, and because you have un-done the winder's clever step-up.
> why it makes a difference if the transformer is teeny weeny
This requires an appreciation of Dimensional Analysis, and it is really too late to go into. However.....
Consider two transformers:
#1 -- 1"x1"x1"
#2 -- 10"x10"x10"
Surface area:
#1 -- 6 square inches
#2 -- 600 square inches
Cubic volume:
#1 -- 1 cubic inch
#2 -- 1,000 cubic inches
Power capacity goes roughly by Weight, and weight goes by cubic capacity. #2 should be good for 1,000 times as many Watts.
However we have losses which means Heat and we shed heat with Surface Area. #2 can only dissipate 100 times the heat of #1.
Finally: transformers always cost too much, so we always sharp-pencil then as small (cheap) as possible.
#1 has plenty of surface area, won't over-heat, even if we use extra thin wire and allow large losses.
#2 will over-heat easily, we have to size the wire to keep losses down just so it does not smoke.
Here's a graph from a book which goes way-too-deep into details. There's a lot going on here, print it and stare at it a while.
Across the bottom is transformer capacity in VA (Watts, near enuff).
Up the side is Regulation, 1% (super stiff) to 100% super-saggy).
Diagonal lines are Heat Rise. 50 deg C is a good number for insulation breakdown. We can pay much more for higher-temp stuff, slightly less for wimpy stuff.
I have drawn a "COOK" line near 50 deg C. Pick your VA/watts on the bottom, read up to 50 deg C, then read over to the left. This would be the Regulation for a transformer designed to run that hot.
But at 10VA, the Regulation "should" be 50%. This is very saggy! Basically we can't make a small transformer cook itself (surface/volume ratio). Design must be on other basis.
Often we would like "good" regulation, maybe 10% ? But below 100VA we have to add material to get good regulation, even though the heat is fine. Much below 100VA it gets silly and we find that a 20% regulation is a better buy, even though we may have to up-rate some of the parts of the load for the uncertain voltage. Down around 10VA we either buy much bigger than we need or we accept "terrible" regulation. Doorbell transformers can be 40% high when un-loaded... the un-connected doorbell does not care, and sizing the door button switch for 40% excess voltage is trivial. With that we also have a part that CAN be shorted more-or-less forever-- it sags so fast that the total heat is less than it can comfortably dissipate even with crap insulation.