Hello,
Sorry to hear you are having issues. While you have had the unit for almost 2 years, I still feel bad when they are any issues with any of the units I have serviced or restored. Did these problems recently start, or if not when did they first occur.
It appears to be 2 different issues in my opinion
Lower frequency static/crackling out of the speakers. Is this apparent on one channel or both???
Usually a noisy transistor will just affect one channel, although I have found situations where I had bad transistors in both channels. The symptom of being the same level of noise regardless of vol control position tells be the problem may be on the main applifier board, maybe the differential amp transistors. And if Differentials act up, they many times cause problems with the DC balance, allowing for DC volts on the speaker terminals, which can cause spotty relay contacts.
The dropout at low volume when cold can be a dirty relay contact. While I always dismantle relays, clean, deox, and polish the contacts, on any receiver I work on, they can get spotty again, although I think I put a new relay in your receiver. Tell me if you see a green LED lit on the relay when energized. And relays can go bad. I don't think it would be a bad coupling cap as I replaced all originals with hi end "Fine Gold" type.
I don't think dirty controls or switches are your issue, because I used multiple types of faderlube on the rotary controls, and remove all lever switches, dismantle them, clean, deox, and polish the contacts, before reassembly, which should give many many years of troublefree operation.
If you can figure out a way of getting the SX-1250 to me, I can see what the problem(s) is(are), and address them