How to tell if capacitors are damaged?
I doubt there is anything wrong with those capacitors, at least based on images you provided. If you want to troubleshoot your boards, remove all power from the board first and then take out your multimeter and start probing. 

Measure across both sides of the capacitors in resistance (or continuity) mode to see if it is shorted. A short would be an indication that either the capacitor is defective or something else on that line (upstream or downstream) is defective. If you have access to an LCR meter, you could desolder the component and test it directly. However, without schematics, you won't know what the value should be. Still, if you get some wacky reading, like a few pF or it shows high ESR, that may be a sign of problems.

When dealing with an anonymous board like this, I like to check the capacitors, diodes and transistors/FETs. Those are the easiest to check and you may get lucky.

There's not much more we can do unless you can point us to some schematics.