Who knows how many kinds of patent applications there are? Can we improve other people's products on the original basis? Can I apply for a patent in my own name after improvement?

It can be improved on the basis of others and used as its own patent. In fact, a large number of patents are improved patents;

the improved patent belongs to the "subordinate patent" of the basic patent. If you implement this patent, it will inevitably infringe the patent right of the original obligee, but if the original obligee wants to implement your patent, it will inevitably infringe your patent right, so you can allow each other to implement the patented technology on the premise of implementing the other's patent.