Welcome to forums :-) They can get narky and rude at times, but this one is pretty ok. Just ignore a**wipes
For a 5A op at 12VDC, your filter cap is 1,000uF + 1,000uF per Amp = ~ 6,000uF, so app 8-10,000uF is the closest ie 2x 4,700uF
At that your inrush / charge current is ~ 2X+ the max op current. Worst case is if you switch on at full load or short.
Bridges are cheap, so use a MIN of 10A, 20A ones are not a lot dearer, and it gives you headroom if you want to go up a bit.
For the variable 1A PS, since you are dropping up to the entire range at min OP, you would either use a BIG heatsink with cooling, or go for the LM338T.
It still needs a heatsink though. Again, it gives you headroom.
Welcome to the forum, you should be safe here :-)