is it possible to install desktop and web modes separately on one computer and use it as per needs?

I'm guessing what you are trying to do is only use one of the licenses at a time on a given computer? If so in TestComplete go to File -> Install Extensions... -> from the menu you can deselect either the desktop or web module. This will release the license of whichever module you deselected. To switch simply go back in and select the module and deselect the other one

You won't be able to install two instances of TestComplete (one with Desktop one with Web) if that's what you're asking. Hope this helps.

If I understand your question correctly - I don't believe so.

The web testing is part of the same installation.