> Should i sell this and invest in a new digital oscilloscope? I like the old stuff much more.
No. Sometime you might want some of the features of a digital oscilloscope but the analogue one is much better to start with if you really want to understand what you're doing. And I can think of nothing that a digital does which is especially useful for audio. Cheap ones are particularly bad as they have a greater chance of misleading you than an expensive one.
Continue with what you have. Use both of them and understand the advantages of the better one (and the limitations of the poorer one).
Don't worry about your probes. They won't affect anything unless you're working above 100MHz and maybe even not then. But do adjust them properly, with a square wave source such as the CAL pin. If you can't do that correctly they aren't suitable for your scope, though they will do no harm - they'll just mislead you with inaccurate measurements.
I would want to have the scope earthed as I would any device that I handle that has a metal chassis. But if the practice in your country is to not earth a device such as an electric iron (or indeed a soldering iron), the scope is no more dangerous than those. At least not to you. It might hurt a more sensitive electronic circuit .. though that also might not be earthed.
I'm sure some philips scopes of that era I've used in the past had a 2-core power cable and no expectation of being earthed. I wouldn't expect to add an earth cable to such a device. But if it is fitted with a 3-core cable, the best practice is to connect them all according to your local regulations. An isolation transformer will slightly improve things if that's the local safe practice. A variac is not desirable but I don't know your local conditions. Maybe it's essential to get the mains into the right range, but would worry about the possibility of it giving you too much voltage (eg if you set it high because local mains is low, but then conditions change and it is found to be too high). If that's the case a constant-voltage transformer or some sort of electronic line conditioner might be a good idea. Electronic equipment is not as tolerant as electric lights, heaters etc.
Since you already have a digital scope (or, possibly, one that can be used either way) take some time to understand how they distort the measuremets. 250MSa/s means that it will display very surprising result above 125MHz and will show some oddities above about 25MHz. In fact, a brand new one suffers just the same problem but it's harder to see. Therefore, yours will instruct you better than a new one ! Read about 'aliasing' and 'undersampling' and try to not just understand it but see it in your own experience.