Does non-matching impedance of amplifier and speaker distort the sound?
You can simply forget about impedance matching for home Audio.
Impedance matching is needed only where the wavelength of the signal comes close to the length of the cable transporting that signal. Electrical signals travel with almost the speed of light through cables, for the highest audio frequency (giving the shortest wavelength) the wavelength is about 15 km. I'm guessing that your cables aren't that long.
Impedance matching is needed to prevent signals reflecting and distorting them. This is usually only relevant for high frequency signals, not audio (exception: analog telephone lines).
In my opinion the "impedance matching" for audio amplifiers is really better understood as: "Can this amplifier drive this speaker?"
Example: some amplifiers are only suitable for 4 and 8 ohm speakers. Using it with 2 ohm speakers (or two 4 ohm speakers in parallel) can give issues.
For headphones this is almost never an issue unless the impedance of the headphone is very low (less than 10 ohms) or very high (600 ohms). And even then, if there is a "mismatch" the maximum volume might be reduced.
Usually home audio amplifiers drive the headphone output from the speaker output via series resistors to give a bit of protection against overloading the headphones as they need a lot less power than the speakers. Because of this almost any headphone can be driver from a home audio amplifier.
Mobile devices running on batteries cannot deliver so much power and voltage so overloading is less of an issue. Since the output voltage on these devices is limited I recommend using low impedance headphones, 30 or 50 ohms would be a good choice.
In either case, you do not have to worry about impedance matching, it is really a non-issue for headphones.
Sidenote:
For speakers the output impedance of the amplifier is relevant. The usual recommendation is that the amplifier needs a low output impedance. The lower the better as that will give it better "control" over the speaker. This is not impedance matching, it is actually a "best mismatch" situation as amplifier output impedance ( < 0.1 ohms) and speaker impedance (> 4 ohms) are not the same.
You can certainly forget about "impedance matching" for any reasonable quality amplifier, because the impedances are intentionally not matched!
The output impedance of a good quality audio amp should be very low - typically something like 0.01 or even 0.001 ohms. The impedance of most loudspeakers or headphones varies a lot with frequency, but is always several orders of magnitude higher than the amp's output impedance.
The consequence is that if the speaker is designed to respond to the signal voltage (not current), the voltage delivered by the amp will not depend on the impedance variations of any particular make of loudspeaker, and the audio response won't have any unwanted peaks or troughs at particular frequencies in the audio range.
Impedance matching only matters if you need to get the maximum possible power transfer from one device to another, but that is irrelevant for designing an audio amp. (But if you were trying to send a signal down a few thousand miles of undersea cable, it would be very relevant!)
The reason the amp specification mentions the acceptable output impedance range for the speakers or phones is simply to ensure that when you turn the amp volume control up to the maximum, (1) if the speaker impedance is too high, the sound level will be lower than what you might expect, and (2) if the impedance is too low, you will be attempting to draw too much current from the amp, which might cause some distortion, and (more likely) will blow a fuse somewhere in the amp to prevent it being overloaded.
Historical note: the above applies to modern amp designs - both solid state and tubes (valves). Some old (1950s or 60s) designs of tube amps were sensitive to the speaker output impedance connected to them, and had a switch on the back panel to select the actual impedance being used (usually either 8 or 16 ohms in those days, though high powered modern speakers often have lower impedance like 4 or 2 ohms). Operating such a "vintage" amp with no loudspeaker connected could in some cases damage the amp - but modern designs of tube amp don't have that problem.
The "< 30 Ohms" designation on your amplifier is most likely not the output impendance of the amp, but the impendance of the load it is designed for. Good audio amplifiers have output impendances much below one Ohm.
Such labeling is common, because impendance matching is not a thing in audio equipment. However, driving speakers or headphones with impendance outside the amplifier's design area might affect sound.
Driving headphones with impendance higher than amplifier's designed driven impendance is likely fine. I would believe the only drawback is that the amplifier can not deliver its full output power since it is not able to generate high-enough output voltages, and you'll need to stay at fairly low output levels. Using the volume knob may be fiddly, but amount of power is not really a concern as headphones never need much power.