Resistor arrangement on LM317's does NOT adjust voltages as shown.
Refer to data sheet (carefully) - you need a variable resistor to local ground.
You can only use a second positive supply to form a negative one in the way that you are proposing if the "energy sources" are isolated relative to each other. eg two unconnected windings on the same transformer. The input of the "-ve" supply will be above ground voltage during operation.
To use a single transformer winding or non isolated sources what you want is to use an eg LM337 which is a less well known negative regulator equivalent to the LM317.
Here is a dual +ve & -ve supply circuit diagram using an LM317 and LM337. They use centre tapped transformer and a single bridge rectifier - but you could use the 2 separate windings as you propose and join them appropriately.
LM337 data sheet here
Add a small "spreading" resistor between rectifier output and 10 mF filter cap to widen the conduction angle. This reduces peak currents substantially and reduces RF noise generated by diodes and gives diodes a far less-hard time on conduction peaks.
I'd like to solder the smaller 500mA transformer onto the lugs of the larger transformer, is this safe/do I need to care about polarity?
Slightly unclear what is intended. Mechanically not wise.
Electrically - use wires.
Is it fine if I stack the two transformers on top of each other?
Probably. Minimal flux interactions with closed cores. Cooling may suffer slightly but easily judged and varies case by case.
If I had both outputs connected in series, would I be limited to 1A due to the second transformer only providing 500mA?
When done "properly" each supply can return its own Imax to ground. A current that starts at V+ and ends at V- will be limited by the smallest current capability.
I don't need that much current but I'm just curious if I'd run into trouble since the transformers are unbalanced. I have a 9V 15VA transformer as well, would that be better matched?
As above. As specified you can have 1A V+ to ground, 500 mA V- to ground or 500 mA V+ to V-.
What amperage of fuse should I use? I'm worried about magic smoke....
LM317 is self protecting within limits.
1A fuse fast blow blows at about 2A in moderate time.
How damaging would a shorted/max current output be on this circuit?
As above.
is there any power loss?(compare to step up transformer,actually I see
there is some thin and thick copper wire binding in the transformer.
Will this affect the power loss?
Primary and secondary share the same core so, the main thing to consider is whether the core saturates more when driven backwards. Here's a sort of worked example: -
If the primary winding is 10 henries inductance, on 240V AC supply the current will be: -
\$\dfrac{240V}{2\pi f L}\$ = 76 mA.
Let's say the transformer had 1000 turns on the primary so ampere turns are 76.
The 12V secondary has one twentieth of the primary turns i.e. 50 and its inductance will be \$20^2\$ times smaller at 25 mH. Now, if you applied 12V AC to the secondary you'd get a magnetization current of: -
\$\dfrac{12V}{2\pi f L}\$ = 1528 mA.
Ampere turns are 50 x 1.528 = 76 (i.e. just the same)
Best Answer
I have some observations and a question.
1) input power is 25 Vac, 5A. That works out to 125 VA.
2) Transformer #1 is 220 Vac, 0.5A. That works out to 110 VA.
3) Transformer #2 is 6 Vac, 20A. That works out to 120 VA.
4) Your desired output is 250V 20A. That works out to 5000 VA or 5 kVA.
As you can see, the numbers between input and desired output don't match up. What you are asking for is not possible.
Now for my question:
1) Exactly HOW are you arranging things so that only one transformer gets input current alternatively?