Install photovoltaic power storage yes or no?

Not really so far. Tesla is planning a recycling facility in the Gigafactory. It will take a few more years until the first generation of mass-market electric cars is done.

It was originally about the claim that the development of drive batteries almost as a by-product results in cheap home storage systems.

Yes, that is definitely the case. The economies of scale due to increasing production numbers will significantly reduce the cost of home batteries with identical technology.

Yes and no. Economies of scale in production certainly exist, but on the raw material side, the whole thing is complicated. Countries like the Congo are also involved, which are not necessarily predestined for the necessary large investments. Of course, it is possible that something will still happen in chemistry and everything will look completely different again. A strongly increased demand (through subsidies or compulsion) also works against the reduction in costs. I expect economies of scale for home storage rather through the introduction of grid storage. They have more or less the same requirements, just much larger. And they will have to come on a large scale if coal-fired power plants are to be shut down in about 15 years and everything is to be replaced by wind and solar.

Of course, it may be that the chemistry will still change and everything will look completely different again

That's true. Cobalt is already no longer needed (Congo) and lithium deposits exist worldwide.

I expect economies of scale for home storage rather through the introduction of grid storage.

I don't, these are used more for grid stabilization than for supply.

And they will have to come on a large scale when coal power plants are shut down in about 15 years and everything is to be replaced by wind and solar.

A few technologies are still missing in the list.
For example, hydropower is maintained and kept ready in NRW, but it is not connected to the grid. Gas power plants exist relatively clean and exceptionally modern with high capacity, but they are currently also not connected to the grid. Biogas is suitable for buffering due to very flexible production. Redox flow systems are becoming more prevalent. Methanized Power to Gas is another storage option to watch – despite efficiency problems. Thinking of electricity without heat will soon belong to the past. We gain efficiency through sector coupling and in the future possibly with large seasonal heat storage from water or metal. Lumenion is interesting in this regard.

The braking of further alternatives is due to price finding via the current Renewable Energy Sources Act on the electricity market. Anyone interested can take a look under the term Renewable Energy Sources Act paradox on the big video platform.

I’m curious about that. The reason for shutting down coal is to save CO2, so it makes little sense to burn gas. Whether there will be enough "bio"gas remains to be seen. Hydropower can only be expanded marginally, the days when cheap electricity was generated simply by flooding Alpine valleys are definitely over. And sure, efficiency improvements can still yield a lot, but in the future we will also consume more electricity due to heat pumps instead of oil heating, electric cars, and all the millions of small technical helpers. I am still very skeptical whether it will all really work out as it should, especially since – as you already mentioned – political wishful thinking often dominates technical expertise.

Gas is nearly climate-neutral in Power to Gas. Bio-gas is problematic. Hydroelectric power plants remain unused and are kept operational. They are simply not used because, under the current pricing mechanism, they are not prioritized and lag behind nuclear power and lignite. We will do well with pioneering technologies, which we will exceptionally also use ourselves. In terms of supply and also economically.