Does the water come in at the right on top, or the right on the bottom, or both?

Both. cold water comes from top right, hot from bottom right.

I would love it if you would explain more. You have unmetered cold water, metered hot water, and you mix them before feeding your fixtures where I assume you feed both the cold and the mixed water and can mix them some more? I have never seen that. What country are you in? What is the source of the hot and cold water? Are they completely independent?

The "mixer" is the white, knob looking thing on the lower T? It can be set to guarantee 100% that no water will escape upwards from the Hot water supply side? i.e. it can be used as a shutoff valve? Can you get us a close up of the end of that mixing valve (showing any labeling that may be there), as well as a close up of the face of the other round device on the lower supply line (which looks a lot like a small meter)?

@jay613 The meter for cold water is in the shaft outside the apartment. Fixtures receive either cold water (does not go through the mixer, it leaves through the top left splitter) of mixed cold & hot water (it leaves through the bottom left splitter). Extra mixing is done in the taps. This is in Europe, Poland specifically.

Is the mixer thermostatic? Perhaps a safety feature required by law if you have hot water supplied from outside your home?

New problem: Now I understand what's going on .... the rusty T is directly connected to the main cold water shutoff. That's another source of stress. Is there another shutoff outside (at the meter?) in case you cannot separate the tee from the inside shutoff?

@jay613 If I remember that correctly, there is also additional cold water shutoff in the shaft outside of the apartment.

@jay613 Is the mixer thermostatic? Unfortunately, I don't know.

@FreeMan I've edited the post with three additional pictures. The mixer has only four markings I can identify: "3/4" next to the know, "-" on the top end, "+" on the bottom end, and "M" on the left side end. The round device with red-colored elements is a hot water meter.

If you turn the white knob all the way to "+", turn off the hot water supply valve, and turn on a hot faucet in the bathroom, does cold water come out? (I might have this backwards, maybe it's all the way to "-" but I think "+" means "more hot")

I'd do an experiment. Turn the mixer knob all the way to the - direction. Turn a hot water tap on somewhere in the house and hold a thermometer in the water stream. Wait for the temp to stabilize and make note of it. Next, turn the cold water shutoff to the off position. Repeat the temp test at the same tap. If the two temps are the same, then it would appear that the mixer can fully shut off the cold water. If they're different (and the cold water tap off is hotter), then the mixer will not shut off the cold 100%. If the cold water valve value is hotter, you've done it wrong do over.

heh, @jay613 and I posted at nearly the same time. My assumption was that - was the minimum setting for cold water mixing into the hot water stream, but I may have assumed incorrectly. If the test above gives wildly different readings, go all the way to the + direction and try again. Also, before you start the test, make a mark on the knob and somewhere on the valve base so you can easily return to very close to the current setting.

FreeMan the more I think about this the more I wonder why either of us thought we know what + or - meant :). Wojciech I added high level instructions to my answer.

I would also suggest taking a wire wheel and cleaning up all the corrosion/mineral build up on the outside of all the pipes & fittings. Use a wire brush where you can't get the wheel in. Since this is exposed, that'll make it look nicer, plus, you'll get a better view of exactly what's going on, and it'll make it more obvious more quickly if you've got a leak in the future since you'll see it building up again on nice, clean pipes instead of having to detect additional build up on already messy pipes.

I tried to adjust the mixer but the knob wont move. So I did another simpler experiment, which answers how does the mixer work: I shut down cold water and opened one of the taps. The result was no cold water running. This means that flow of cold water to the mixer is unidirectional.

I accepted @jay613's answer but FreeMan's comments were also informative and added a lot to the discussion.

Your last pic with an alternative layout is interesting. Post it as its own question to get further input on that, since it really changes the meaning of this question and there really wouldn't be one good answer to both questions.

This is the most asinine installation I've ever seen.

If you want more freedom of movement (alignment headaches) replace the tee with a straight piece of plastic pipe, replace the cold manifold with a 5-outlet one, and use one of the outlets together with plastic pipe and whatever bends you like to use to feed cold to the mixer.

Also at the end of the cold manifold the last pipe goes into a red conduit. Maybe the color is meaningless but does that possibly feed your hot water heater? If so, the experiment where you turned off the cold supply to check the mixer means nothing.

your suggested alignment fix probably wont fit in the room available, you could consider using a union between the mixer and the tee. that probably will fit.