Yes you could swap your HD between your systems as the HD interface (SATA) is common to both, But... Your OS versions you have on these systems now may not work correctly when you switch the drives as the CPU's in the MacBook may not support what your MBP has on it.
And yes you could get a USB to SATA cable (if you don't already) to attach your drive directly onto the new Mac that you're getting and then either manually copy over your files and apps or clone them over to the new systems HD. The iFixIt kit offers the needed hardware & software for the task [product|IF107-108|SATA to USB Kit]