I'm fairly new to Krita but experienced Photoshop user. Since I'm using equally Windows and Linux, I'm using Krita more and more.
Every time when I start Krita, the Brush tool is always pre-selected. Can I change this behaviour?

Sorry, no. Krita is a painting application, and in our experience, remembering the last used tool throws off that part of our users who aren't experienced with photoshop, so on restarting Krita, it's always the paint tool that is selected.