1) Like I said, your boombox does not have a conventional line-in feature. If it did,you would have that option available in your function selector and you would be able to play that just like as if you selected "tuner." That you need to depress the record button indicates that you are monitoring the line-in during a recording session, unless you tell me you have a "special" record button lever that allows you to operate the record button without the deck working or without the need to simultaneously record-pause. In any event, it is irrelevant to the issue at hand. The only thing that is important is that you have this issue and it appears related to your tape section. Or... is it possible that it is also happening during tuner mode too but due to a higher signal to noise condition, you are not hearing it? Please tune your radio to a quiet location on the bandwidth and see if is still making the noise?
2) Reflow does indeed mean to remelt the solder/tin. However, you need to observe and recognize if the connection is a good solder joint or not. This takes some experience to recognize whether the parts to be joined are wetting or balling the solder. If the solder is balling, then I would desolder the entire connection by removing old solder and doing it over to ensure that the solder is actually sticking. A bunch of solder balled around a lead is not what you want to see, although I suspect from your posts here that you have sufficient experience to know the difference.
3) Quick test is exactly that.... it's quick but since the results aren't always conclusive, what you are looking for are numbers that make no sense and obviously wrong. For example, you never want to see capacitors or resistors shorted or open although resistors usually open rather than short. You never want to see resistors that tests higher in-circuit than the printed value since that is almost a guarantee that there is an issue. Through experience, you'll learn to recognize that low value resistor readings can generally be trusted more than high value resistor measurements and when they are off, usually a quick examination of the circuit diagram tells you whether that off-spec reading makes sense or not. You don't really need to "compute" the actual expected value since you can usually see at a glance whether a reading is about right based on the circuit diagram. The "quick" test is something we do simply to see if there is something obvious.
4) How "off" is "off." There is almost always some amount of deviation, not only in the circuit itself but also in the equipment you use to do the testing. The multimeter itself has some inherent inaccuracies and that's before you add the probes and the surface oxidation of the parts tested. Resistor drifting really isn't critical. In fact, if you have a 1k ohm signal resistor in the auido circuit and it has now changed to 10k, your boombox will likely still operate sort of normal although the audio might be a bit attenuated. However, an 100R resistor that has changed to 130R in a bias circuit could cause enough problems to become noticeable or problematic. V/R (voltage/resistance) measurement tests are useful to see if something in a circuit is amiss since it gives us a good picture of the overall health of a circuit, but the V readings posted usually is only accurate when the power supply is at spec. So if the schematic shows 12V at the power supply and it is currently powered by AC or batteries, then the voltage could be different and the voltages in any non-regulated section will all be globally off.
5) A 68pf capacitor will charge almost instantaneously. Any small capacitor will charge fast enough that, depending upon the voltage applied to the circuit under test by the particular meter, be too fast to see. A resistance test on capacitors isn't a very good way to test them anyhow unless they are either shorted or open, and with small capacitors, even that is usually inconclusive. This is where a higher end DMM is useful since they usually have a cap testing function, or an LCR meter. Some of the cheap transistor testers I mentioned earlier can test small caps accurately. Out of curiosity, I just now stuck a brand new 82pf ceramic cap into one of those testers I got off eBay and after about 3 seconds, the reading came back at 82pf.
6) Replacing all electrolytic caps is certainly an option. Ideally, I'd like to see where the problem is first, but lacking some more sophisticated instruments than a cheaper DMM, then it's certainly a service that one could try. Some folks don't like the shotgun capacitor approach but if you ever want to recap a boombox someday, might as well be now since a partial replacement today means that in the future, you will end up replacing the same ones again? Would I recap a boombox like this? Absolutely without a second thought. The service manual only shows 12 transistors total for this boombox so it's pretty simple and doesn't have that many components. By comparison, the Marantz CRS-4000 boombox, about similar size to this one which I recapped about 10 years ago had around 100 electrolytic caps, if I recall correctly.
7) I don't agree that ceramic caps do not go bad. I have definitely seen them go bad. The difference between electrolytics and non-electrolytics is that electrolytic ones have a more predictable failure rate based on age and usage and heat, and stress whereas non-electrolytic ones fail randomly and therefore are not replaced unless they go bad. That's why you test them if you suspect them.
8) By the way, this is not a very sophisticated boombox. One look at that PCB shows this. Unfortunately, I find that simple designs like this often aren't always as immune to noise as they can/should be.