Firstly, I wouldn't be poking my finger anywhere near the output (or the output transformer) of that circuit.
Secondly, Well done!
Today I finally received the correct and new IC UC3843, and it worked right away

!
Excellent!
At a supply voltage of 10.7V and output 300V this converter draws about 35-40mA unloaded, when an analog voltmeter with a resistance of 270k is connected, the current draw increases to almost 100mA. Any thoughts about, how good or bad that is? I tend to think, that it's not really good, but then again- this is my first working DC booster.
(And that analog multimeter reads 300V?) Well, ignoring the quiescent power consumption of 400mW or so, for an output power of 333mW, the input power is 642mW, so you're looking at about 50% efficiency.
That sounds really poor, but efficiency normally increases with load up to some maximum value before dropping again.
The best thing to do is to measure input and output power for a range of loads, and then graph the efficiency. From your statements below, graphing the output voltage would seem to be a good thing too.
It seems, that my current setup won't be able to deliver promised 150W of output power. Maybe the transformer is too small or the core is from a bad (unknown) material.
Well, the transformer is critical. The correct choice of transformer will depend on the voltages, the frequency, and the power you need. If the transformer is "unknown" it may not be suited to any of these criteria -- although clearly you are getting the voltage you want.
The output voltage of ~250V at a steady supply ~12.6V dropped to some 215-220V, when it was loaded with a 100W 230V rated bulb, the current was ~0.41A, so the output power was about 90W, but the voltage could not be kept at a steady level. This didn't happen, when loaded with just a 40W (maybe 60W- letters not visible) light bulb.
When you say "the voltage could not be kept at a steady level", can you describe more exactly what happened to the voltage, and anything else relevant? Did it drop over time? Didi it fluctuate? Did it fall to zero? Did something start to smoke (or get really hot)? Does the input voltage start to fall?
The transformer makes some annoying low frequency noises (zip archive), mostly when unloaded or slightly loaded. These noises change, when I touch one of the transformer's output pins or its core (should it be conductive?).
It is quite likely that at low powers the transformer is called on to provide intermittent bursts of power and it is not until the power exceeds some value that the regulator needs to be running continuously. What you are probably hearing is a single pulse of (say) 20kHz sound. You hear that as a click. When it's running continuously, the sound is probably a lot louder, but you can't hear it.
The sound comes from the transformer. As energy is stored into it and released, parts can flex and this produces noise. You can damp this out by securing the transformer to something solid in many cases.
I know, that these measurements are not lab-grade, but for a quick test they should be OK.
They're lab grade for your lab, and seem perfectly acceptable at this stage
