So, in the priority that you recommended to me, guys:

1. One guy in a parts store has a tester. He felt relay # 72 - it works (clicks, conducts electricity);
2. Fuses No. 63 and No. 67 visually did not die;
3. I didn’t look at the bulbs - it’s hard to get to them, in the Urals it’s already autumn, it’s raining, cold and muddy outside. But I left that for last, it is unlikely that both lamps will burn out at the same time;
4. I am inclined to believe that the steering column switch is ill. Indirect confirmation - 2 auto electricians independently of each other to my question "What could be the problem?" immediately pointed to the steering column switch.

Here's what I'm planning:
I will look for this steering column switch in Russia. There is a possibility that GM did not invent unique controls for the HHR, but applied a ready-made solution. For example, my radio is identical to the Chevrolet Aveo. Suddenly the switch too - it would be great, I can solve the problem without expensive shipping from the USA.
If it is still unique, I will try to repair or buy a contact group.
Well, buying a part in the USA as a win-win, but an expensive option is always with me.
I'll check the bulbs somewhere in the middle of this path.

 

Keep us updated

 

as always

 

If the blue indicator in the dash comes on, the switch is probably ok.
My guess is bulbs.

 

Seriously?
Well, this statement seems to me sound.
Either way, the bulbs are still scheduled for inspection. I'll do this before I spend the money.
Thank you

 

Six months have passed since that time, and my hands reached (finally!) to the head light bulbs.
And yes! Prod and the other guys were right - poor quality bulbs. It was nicely packaged, it had a nice white light, but the base had to be filed or the bulbs wouldn't fit into the connectors. And the high beam died in both bulbs at the same time.
I don't remember the name of this Chinese manufacturer, but it is still the leader in the production of beautifully packaged sh*t.