Yes, direct or electrical ones are available. The PV based ones are not different than any others from the store. Some examples for the direct solar powered ones below:
Most solar cookers don't store energy. You might be able to make one using a phase change material that stores solar energy at say 200C for long enough to cook something at night. It would be an interesting engineering task!
Yes no problem. Cost may be an issue. A 5kWh storage system, which will give you a couple of hours usage is currently in the vicinty of $8000, I believe. When you consider that is worth about $1.50 @30cents/kWh, it might be easier simply to draw from the grid.
There are solar batteries which have the capacity to store the energy required to power the size (power rating) of the solar stove. If a single size of battery find in the market do not have the required capacity you can combine batteries together to increase their capacities to the required size.
If all you want from your system is to cook things, which requires heat at some modest temperature, I really don't think a PV-battery-electrical stove is going to be very cost effective. Efficiencies (at affordable costs) are about: PV 15%, Battery charger and MPPT 95%, Battery 85%, inverter 95%= 11.5% overall. Then you connect the stove. You can capture heat energy from the sun with at least 50% efficiency so the only difficulty is storage. Your store need only be 23% efficient to match the electrical solution. And it would be far, far less complicated.
I agree with Tony Maine. Cost is an important consideration. One must consider ALL of the three TBL factors Social, Environmental AND Economic. Just because you CAN do something it does not mean you should. There may be other reasons why you would not.
A simple alternative might be to cook while the resource is available - daylight hours.