Can I use the screen (and keyboard) of a separate laptop with a Mac Mini? At the moment, I really need a Mac to run Logic Pro, and I reckon my cheapest option is getting a Mac Mini.

It would be very convienent if I could use the screen (and optionally keyboard) of a current working Windows laptop with the mini.

Can I do this?

(Update maybe like Apple Remote Desktop on Windows?)
There are many VNC applications that could accomplish this, and OSX has VNC support built in.

However, I really think you're better off buying a monitor, mouse, and keyboard for the Mac Mini.  Get something used if necessary.  You don't need to buy them from Apple - any standard monitor, mouse, or keyboard will work fine.