Can a "desktop pet" be made with Unity?

I am confident I can make a virtual pet simulator, but is it possible to have said pet roam around the screen whilst the user browses the web or attempts to work?

The main obstacles I imagine are:

• Creating a transparent game window

• Forcing said window to always stay on top

• Adapting the window boundaries based on how much is visible

• Moving the window to simulate the subject's movement

• Clicking through transparent areas of the window? (say, if a pet and a ball exist and the user wishes to open a desktop shortcut between the two)

I have been searching and brainstorming ways to do something similar to this and thought I would ask here, as Unity would be my preferred development platform by far. If it is impossible in Unity but possible in another language, I would also love to know.

Thanks in advance for any leads!