How does insulation in brick construction affect the indoor climate?

Additional thermal insulation should by now be standard.

What "standard" is, I am not quite sure regarding Austria. That means especially I do not know which equivalent of the "Energy Saving Ordinance 2016" applies there or not.

now I have heard several times that bricks are also very good for the indoor climate because they conduct moisture very well.

Putting aside the naive image of bricks (handmade from free-range lumps of clay by happy cows), they are no longer the only real organic wholegrain bread. The differences between masonry materials are less dramatic than their respective proponents and detractors suggest. Apart from that, the plaster is just as relevant for the indoor climate as the masonry material.

Whether it is really "eco" to carve a house out of a solid tree trunk, I mark with a tree-high question mark :)

But do bricks then retain their good properties?

All the properties that a brick has, it can only directly (or via the plaster and the surrounding air) "transfer" – not to an insulation panel, and not through it. Any specific property practically gets stuck in the non-relation there.

I always have the impression that fresh air from outside must be better than air redirected multiple times through pipes.

Real air is always best "analog," I agree with that ;-)

Controlled residential ventilation and hygiene ... There is no reason to assume that the pipes would systematically and fundamentally become dirty or even microbiologically infested. The conditions simply do not arise. Unhealthy mold infestation in single-family homes is orders of magnitude smaller (practically excluded) when using controlled residential ventilation than in houses without controlled residential ventilation -- mold can certainly occur there if the residents do not pay attention.

All other air parameters are always better with controlled residential ventilation than without, unless windows are left open permanently.

Is the same question actually asked about cars (duct system, filter, air conditioning)? It would have to be extremely unhealthy as well (apart from NOx etc.).

Workshops do offer climate disinfections ;-) Helps against odor.

Are there any objective information about the advantages and disadvantages of [Holzriegel], [Vollholz] and [Ziegel]? On the internet, almost everyone praises the building material they have themselves.

But how is it really with heat in summer, sound insulation and indoor climate?

Enter "Raumakustik und baulicher Schallschutz tu Berlin" into an internet search engine of your choice! Unfortunately, linking is not allowed in this enclave of the WWW. Hopefully, that is objective enough for you ;-).

In general, I find baunetzwissen dot de comparatively well-prepared and IMHO objective.

You can actually find objective articles about "sommerlichen Wärmeschutz" on the web. Just a note: Ziegel is a vague term; there are different kinds...

Room climate... Yes, that might be more difficult to find something factual and objective about. It is more subjective and burdened by myths ("breathing walls").

Otherwise, take a look around your local library!