Is it possible to store static electricity in any type of battery and can they be used in electric appliances (lightbulbs mainly)?
Yes indeed. Assuming that by storing static electricity you mean storing an electric charge directly then there is currently a lot of interest in devices called supercapacitors that do exactly this. In particular they are being investigated for use in electric vehicles.
Storing a high charge density in a capacitor is hard because it produces a very strong electric field. Then you need to worry about dielectric breakdown, leakage, and so on. Supercapacitors use a relatively low charge density but have an enormous surface area, and the result is they can store a great deal of charge in a small volume.
Rubbing a balloon generates a static electric charge. Some electrons are separated of their nuclei. Electrons have a repulsive force on each other. They are located on the spherical shape of the balloon. Uniformly distributed because of their repulsion.
Touching the ballon, these electrons flow - a small electic shock. Moving electrons are called electric current. If current is too low, we build a bigger balloon. So it should be possible to build a battery?
No, because the static electricity does not produce a constant electric current. The balloon is discharged, once touched. Instead of thinking of a battery, I like to think of it as a capacitor. LED lights of a bike sometimes are buffer with one of these. Unlike a battery based on chemical reaction, static electricity in this example has to be recharged. Rubbing the balloon is not practicable.
Have a look at the Van-de-Graaff generator generator. This device constantly generates static electricity. Educating and fun if you touch it.