Yes.
Yes, home heating oil and diesel fuel are almost identical and can be used interchangeably. Home heating oil used in a diesel vehicle is subject to road tax and you can be fined if caught using it to avoid road tax.
Yes, but since biofuel is a substitute for diesel its only used in diesel burning cars.
powering cars
The main fractions of crude oil are gasoline, diesel, jet fuel, and heating oil. Gasoline is used as fuel for cars, diesel is used for heavy-duty vehicles and machinery, jet fuel is used for airplanes, and heating oil is used for heating homes and buildings. Other fractions like asphalt and lubricants are also derived from crude oil for various industrial and commercial applications.
Most cars do not have underfloor heating mats. In fact, underfloor heating mats are not used in cars. They are used in homes to clean floors from being cold in the winter.
No, diesel engine require diesel fuel. Put petrol in a diesel engine and you will destroy it.
There are different grades of diesel fuel and heating oil but in general they have similar qualities including the flash point. The flash point is the temperature at which the fuel ignites on its own, without a spark. Diesel fuel used to only be available along truck routes. Before diesel fuel became easily available everywhere, people with diesel engine cars would use heating fuel, but would have to make sure it is clean, passing it through a filter to remove foreign particles.
vegetable oil
For cars? Diesel and gasoline.
No, they make gasoline powered cars too.
I don't think there is such a thing as a "diesel battery". Perhaps you mean the batteries used in diesel cars; the batteries usually used in cars are lead-acid batteries, the acid being sulphuric acid.