Should it be possible to take a screenshot with the Print Screen button and paste it directly into Inkscape? When I do that nothing happens.

If I press PrtScr or Alt+PrtScr and later Ctrl+V in Inkscape I get the corresponding image in Inkscape. Sorry, no idea why it shouldn't work for you. Do you have some Screenshot Tool installed?

I have no screenshot tool installed. The machine belongs to my university so it might not be configured the way a domestic machine would be.

Edit: I just tried it again and now it's working. I don't really understand because I tried multiple times before posting.

Windows 7, and I assume Windows 10 has a built in tool. I use it, but I've been thinking of searching for a freeware tool with more helpful features. (Particularly, I want to be able to capture the mouse and pointer icons, which neither PrtScr or the built in tool will do.)

You should be able to find it in Windows 7 at Start > All Programs > Accessories > Snipping Tool. I'm not positive Windows 10 has it, but I don't know why it would be removed.

Although I can also paste into Inkscape from the PrtScr button. (using Ctrl + V and as well the Paste button)

One nice thing about the Snipping Tool. You can copy the image you "snip" and paste that into Inkscape (or wherever), and not have to save it first and then import.