Electrical installation - is everything okay with it?

As promised, I took a few more photos, maybe that will help...

Did you always have to replace the same fuse or different ones?

3x 16A is of course not too much for an entire house.

3x 16A with WP and then such a distribution?!
Oh dear.
Let's get a master involved, an electrical master who knows what he's doing.
In my opinion, this should be fundamentally checked and renewed.

3x16A is too little for a normal house. If the heat pump is on and some other things, e.g. in the kitchen, are also running, a 16A fuse trips very quickly. Since the three fuses are your main fuses, the electrician should take another look and check how strong the supply line is. Usually, a 10mm² or 16mm² cable is installed from the house connection to the meter cabinet. These can be protected with 50A or 63A fuses. Since there may be 63A in the house connection, you would have to put 50A in the meter cabinet because of selectivity.

This is how it looks, I only see more loose cables and terminals.

The heat pump is definitely of interest to your energy provider. In any case, we already had to specify that. We also have a line set up so that an electric car would be possible. With a heat pump, it needs to be well distributed. With 3 lines, the circuits and consumers should also be reasonably divided!