When you initiate a search with the K2 Tools module with search function, the url looks like this: 'mydomain/search?searchword=test&categories='. Is it possible to change it so when the search is for all the categories, the categories part wouldn't display? The url would be much simpler that way like only 'mydomain/search?searchword=test'.

Hello,

Unfortunately this cannot be changed.

It is sad news. Not even by heavy modifications? I think it's something you should keep in mind for future releases.