Can I program HV-MS735 gaming mouse on one computer, and then use it on another computer without installing the driver and programming it again?

The settings of the buttons will be stored inside of the mouse, so even if you use the mouse on a different computer, the setting will still remain the same, so you should be able to use your current setting on a different computer. But if you need to program the mouse on a different computer, you’ll need to install the programming software on that computer as well.