that the first modules from Trina Solar with 670 heat pump are supposed to be available in Germany. That is of course a big deal.
All modules over 2sqm may only be installed outdoors in Germany and have no approval for mounting on the roof, in most cases these are max. good 400W.
The storage should be sized to get through the night. It is expected to be charged during the day from April to October, the remaining 4 winter months the storage will mostly not be able to be charged at all, as the generated power can be consumed directly during the day. In the summer months, more power is permanently generated during the day than can be consumed.
Assuming a constant consumption at night of 300W (in my opinion relatively high), 3.6kWh of storage would be sufficient for 12h of bridging time, therefore I would install a 5-6kWh storage in a single-family house.
Regarding e-cars, I do not think "intermediate storage" in the home battery is worthwhile. To store 100km, you would need about 20kWh of storage including losses with the associated costs.
If a heat pump is built, it uses much more in winter anyway than the photovoltaics will supply. Overall, a large battery rarely makes sense when you take the costs into account.