Cabinet in front of insulated exterior wall - risk of mold?

I also see that as critical. Maybe leave out the back panel and ventilate through the doors/top carcass?

A wardrobe is often in the bedroom. Anyone who loves a cold bedroom should definitely not leave the door to the heated living area open during the heating period. Although warm air usually has a relatively low relative humidity in winter, it still contains enough water to form condensation on an unheated bedroom wall. So heat only in the room, but never with (moist) living room air. Then, with sufficiently insulated exterior walls, there will be no mold behind the wardrobe.

The bedroom can be a bit warmer for me, I am less of a typical fan of a cold bedroom. Only what I don’t quite understand: Why is it good to heat the bedroom, but if the warmth comes from other rooms, is it bad?

Do you both have opinions on heating cables on the wall, ventilation grilles in the closet, or maybe also on a chemical anti-mold impregnation of the wall?

If all rooms are equally warm, it doesn't actually matter where the heat comes from. It is only about the fact that in the coldest room, with possible air exchange between the rooms, thus equal absolute humidity, logically the relative humidity is highest. Normally, this does not cause any problems. But there are still people who prefer a cold bedroom. And in this very room there is a wardrobe with many clothes inside. This wardrobe prevents the outer wall from warming up, so that behind the wardrobe it becomes even colder, and in extreme cases the wall is moistened by the cooled indoor air and then molds.
It is simply the case that warm air can contain relatively much moisture, which is quite desirable in living areas, since the relative humidity becomes lower the warmer it is. But this air can be too moist for a cold wall in the cold bedroom behind the well-insulated wardrobe. So never let warm air from the apartment into a cold bedroom with a wardrobe on the outer wall. If it gets too cold, it is better to turn on the heater in the bedroom.

Sorry, now I forgot to answer the last questions. So, I would skip heating cables. I'm too afraid of fire hazard. In normally heated apartments with insulated exterior walls, it's certainly no problem to place a wardrobe against them. I only know mold in houses with old solid brick walls without additional external insulation. I wouldn't use any fungicides against mold either. I lived in an old house until university. The exterior wall of my cold bedroom was still the old city wall to which our house was attached. You could only heat rooms here where a chimney connection was possible. Mold was something completely normal for me as a child and never bothered me. I never had allergies or other unusual illnesses. That's why I might be a bit too relaxed about this topic.