In various manuals (regardles tube or solid state) I read that I should turn amplifier on with volume down. What's reason behind this? Is it protecting my hearing or amplifier itself?
When the power switch for an amplifier is initially turned on, there is an instantaneous surge of voltage and current into the amplifiers power supply. When the power supply circuits become fully charged, which happens almost instantly, that initial surge levels itself out and the amplifier circuitry becomes stable. The initial surge may cause damage to loudspeaker voice coils and the sound of the surge can startle or otherwise cause dismay to unsuspecting listeners. This is usually more of an issue with Solid state electronics, because tube electronics take a short while for the tubes to warm up before they will conduct electricity, but while the tubes (valves) are warming they may still need to stabilize before they are ready to function correctly, and while they are stabilizing, they may oscillate, causing strange noises to be heard from the speakers. All these situations can be avoided by turning the master volume control down when powering up for performance. I hope this helps you understand the reason this should be done.
Depending on what amp, and what's plugged into it, when the master volume is at 0 there is nothing at all going to be amplified. Master is most important, as it controls everything on the amp. There will sometimes be a sharp noise otherwise, as the eletricity surges over the contacts of the switch, which will possibly cause damage not to the amp, but the speaker/s it runs through.
There may also be a mic or guitar connected, which, if the amp's volume is up, may cause feedback which at very least is annoying, at worst will harm said speakers. Were those sounds loud enough, close enough to your ears, some harm to hearing may occur, but it's unlikely.
All that apart, pros will always fire up that way - you can often tell it's an amateur band with all the bangs and pops going on - let alone the blowing into or banging the mics to check them...
The fact that you suggest it might be your hearing that is being protected tells us you already know that unwanted noises can occur at switch-on. (To which we could add at switch-off, when plugging or unplugging a cable and the possibility of feedback.
None of these are likely to damage the amplifier as such. They might damage a speaker connected to it. And, yes, they might not do your ears any good (although ears are pretty tolerant of short-term shock, it's prolonged 'standing in front of the speakers' that tends to cause the real damage).
Anyway, for all these reasons, it's standard advice to turn the power amp down before switching on, and it's why powering up your entire studio through one master switch is generally not a good idea.