Photovoltaic system battery storage subsidy feed-in tariff?

Yes, that is one of the questions - with a 3 kWh battery (which would probably be economically optimal) I could theoretically get by almost completely without drawing electricity for 5 months and achieve 300 full cycles (based on an annual evaluation). Nevertheless, it is only worthwhile if the battery would cost under 2000€, unfortunately it costs at least 5, without installation.

I will soon measure myself; I do not have access to the friend's measurement. Some simply assume that the storage is only used at night. But this is not the case. Whenever the sun delivers less energy than is currently "requested," the hybrid inverter will take energy from the storage and not from the grid. This is why there are always charging and discharging times of the battery during the day as well.

Yes, that is one of the questions - with a 3 kWh battery (which would probably be economically optimal) I could theoretically get by almost entirely without drawing power for 5 months and achieve 300 full cycles (based on an annual evaluation).

No, they are not that expensive anymore. Very solid and with a good BMS are, for example, Pylontec batteries, which cost just under €500 for 1 kWh of usable capacity. US 2000 Plus, for example.

I am now building the battery slightly smaller and have reduced it from 12 to 9, I can still upgrade it. It runs on 3 phases to also be able to use the battery power for [vehicle], [sauna], [instant water heater], etc.

It's not just the battery - the control system is also required, which is more expensive than the battery. I would also like to have a battery if it would at least somewhat pay off. I have SolarEdge, to integrate a battery here I first need a StoreEdge and a single-phase inverter and 2 control modules, those 4 things already cost 2500€. Other providers might be somewhat cheaper, but not by much.

Yes, you ideally control batteries with a low-loss hybrid inverter. Depending on the maximum power of the roof, they start at about €1100 even from reputable manufacturers such as Kostal or Delios.