Is the thermal insulation facade damaged by nails in the exterior wall?

Well, that's not true for Danwood – the household manual even explicitly and thoroughly explains how to attach (light) objects to ETICS.

What is not allowed, however, is to fasten heavy objects (such as awnings or similar). But this has nothing to do with the insulation; rather, it specifically concerns the wooden studs. If something like that is desired, Danwood plans additional reinforcements (beams), and the fastening position is accordingly predetermined. Depending on the mailbox, such a reinforcement may also have been planned there.

It also depends on the WDVS, one made of EPS easily forgives small nails, one made of wood fiber insulation boards does not! If my tenant were to put nails into the WDVS made of wood fiber insulation boards, I would be seriously upset, that is not okay! Then, of course, close the holes with cartridge sealant (permanently elastic) - done.

I also strongly assume that the gutter is not attached with nails but with screws (if on/in the WDVS). Usually, gutters are attached to wood beneath the last layer of tiles.

There are special plastic anchors for WDVS. I have used such for lamps, etc., and even additionally coated them with cartridge sealant before screwing in the last millimeter so that they remain completely and permanently sealed.

A WDVS made of wood fiber insulation boards can rot happily behind the exterior plaster for years without you noticing. The damage can become very, very large, up to the replacement of the affected wall!

I read an incredible number of contradictions here, combined with concessions, but also denials of guilt.

I'm not saying that it was okay on our part. The question now is whether something like this can seriously damage the thermal insulation facade.

Yes! At 2 cm damage, the plaster is penetrated and no longer does what it is supposed to do.

It's funny that the landlord attached a gutter right next to our little nails, which of course is attached with much bigger nails. He probably didn't worry about the thermal insulation there :)

Check again: are those screws or nails? Nails don’t hold a gutter.

It should be added that the landlord has wanted to sell the house for a long time, and since that would surely be easier with an unoccupied house than with an occupied one, he is trying to get us out by any means.

You won't get a termination for property damage. The liability insurance will pay for that anyway… this mention at most shows lack of insight.

My common sense actually says that this can't be the case,

… naivety can rather be explained by ignorance, but not by sense.

As far as I see it, something like that should be fairly easy to repair again (since the nails hardly go into the wall) or do you see it differently?

The nails damaged the 2-3 mm plaster. Plaster serves the airtightness of the house. Whether 2 nails or a paving stone: damage is damage.

Aha. From what diameter of stone chip would it no longer be acceptable if my daughter throws pebbles at other people's cars? And up to what point would the owner have to tolerate it because the metal underneath is not damaged?

EPS, wood fiber, Styrofoam, glass wool. Either way... the OP gave his landlord an easy opportunity. Because before you start putting holes in the facade, you (unfortunately?) basically have to ask the landlord. And an expert will always play it safe in such a case and say, "Well, you can't 100% rule out long-term damage."

You simply never can.

I find it questionable to want to accuse the landlord of overreacting now. It is his property...

His property, yes. But it makes a difference whether the car came into contact with a boulder or if there is a polishable scratch in the clear coat. For the latter, you cannot demand a new car. Sometimes you have to keep things in perspective.

I read an incredible number of contradictions here

You can also overdo it. Please name just one concrete contradiction!?

You won’t get a termination over property damage. That’s covered by liability insurance anyway… this mention at most shows stubbornness.

In what way is that supposed to show stubbornness?? We have effectively received the termination. I haven’t taken the time here to go through the entire history between us and our landlord because I wanted to focus on the one issue with the facade. The fact is that the landlord has wanted to sell the house for some time now and wants to get rid of us by any means. He hasn’t even properly looked at the "damage" (which is on the facade on the 1st floor) but saw it from afar and immediately added his termination because of it. He neither sought a conversation with us nor issued a warning.

… Naivety can be explained more by ignorance, but not by reason.

Very philosophical and helpful...

The nails damaged the 2-3 mm plaster. Plaster serves to make the house watertight. Whether it’s 2 nails or a brick: damaged is damaged.

damaged is not damaged. The question is also how complex a repair is… whether it can be done with simple means or would be more elaborate.