The only thing that I will say here is you get what you pay for. FSP's should be decent for the money. Corsair is of course a great choice but it's one of the more expensive on your list. I can get corsairs cheaper then that at my local microcenter but I am not sure where you live.

1) You should be good on no -5v rail. AWE64 doesn't use it.
2) Amperage is a rating keep in mind. Quality PSUs tend to go higher then they are rated, and cheap PSU's tend to be overrated. 16a should be fine. The +5 volt being weak doesn't useally come into play unless your running a high end SlotA/Socket A system with a power hungry AGP card. Then you might need 20 or 30amps. A Pentium MMX should be fine with even 10a.
3) the +4 connector should be something you can split off on most ATX power supplies that I have seen.

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Regarding PSUs, Dell and HP often feature excellent quality supplies from the likes of Lite-On or Delta. They're generally extremely well constructed, fairly well available due to extensive use in OEM designs, and are usually not expensive. The downside is that they're not always the most powerful, but can work very well for more modest retro builds like yours.

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Where are you located? A good Seasonic can be gotten for less than $20 shipped if you are in the USA.

Those caps look okay, visually. Of course, some caps don't fail visually while still being well out of spec, so it's not an absolute guarantee that they're good. The junk underneath the main bulk caps is most likely caulk that has degraded with time. You can test for any shorts by jumpering the PSU alive, crossing the PWR_ON pin (has a green wire) and one of the GND pins (with black wires) on the ATX connector without a load (or with an appropriate load resistor) to check for any arcing.