Coil whine is not a sign of anything really. Switching PSUs use various techniques to regulate a high input voltage to a low output voltage. Basically, they allow the high voltage to pass through in short bursts, and then through passive components that form a sort of low-pass filter, those short bursts of high voltage get averaged out into a steady but much lower voltage. When the PSU is off, there's very little load on it, so it takes less of an "on" burst to keep that steady output going. Some PSUs achieve this by keeping a regular interval of bursts, but scaling back how long the burst is -- the venerable pulse width modulation scheme, or PWM. Others keep the pulse WIDTH the same but pass that burst through less often -- modulating frequency instead of pulse width. In those cases, the frequency of pulses can start to dip down into the audible frequency range -- maybe 10 to 15kHz versus 50kHz, or 100kHz, or even up into the MHz (though not typically for this kind of PSU.) When it's ~10-15kHz, you'll hear the whine. Just like you do in a TV set where the yoke deflection coils are humming along at 15kHz raster line scanning frequency.
That said, sometimes that coil whine is indeed because of lower pulse frequency, but because the PSU has gone into current-limit or other protection mode, and so it's trying to start up, detecting a fault, and turning off again. Then it retries some milliseconds later. Obviously, if your PSU is working properly, this probably isn't the case. It's just a long way of saying "it depends."
The boxed film caps are the blue and white ones near the large coils in the side shot you took. Those are pretty reliable, so you should be OK for a while. The other caps look healthy as well. Impossible to say for sure, but if the PC seems stable, you can make a judgement call on whether to pursue cap replacement now, or only when/if you have trouble with it.
Unfortunately there's just no such thing as insurance against damage to downstream components. A PSU will ideally fail gracefully, and protect the downstream load at all costs. But sometimes you're just not that lucky. All you can do is try to use a quality PSU and hope for the best. There's just no getting around the risk entirely. This one seems like it's built pretty well, so I would not hesitate to use it.
A multimeter is nice, so you can see the output voltages, but you really should be loading the supply before you measure it anyway. An unloaded supply doesn't necessarily look safe to use. I have some quality Delta supplies that show wildly wrong 12V and 5V rails when run wide open. And they will only allow me to do that for a minute or so, then they go into protect mode. Attach a motherboard and they'll settle down and run like champs. A multimeter also won't tell you what the DC really looks like. It'll give you an average (maybe even RMS) voltage, but it won't necessarily tell you there's 4V of ripple on your 12V line. You need an oscilloscope to get the full picture.
That is to say, unless you have a sacrificial test jig of some sort, eventually you just have to hold your breath, plug it in, turn it on, and hope for the best. Maybe put in a cheap video card at first, and save your OG Sound Blaster 2.0 and Voodoo 5 for when you have more confidence in the supply.