See if the Disk is issuing any S.M.A.R.T errors in Disk Utility...
To be more clear: Could a memory shortage mess up basic functions like drag & drop and cut & paste - even when the application in question is not open or active?
Yes, if 'memory' is defined as free space on your hard drive. (Memory is actually RAM).
Drag/drop relies on swap files etc and needs space to breathe! Too little free space and your operations will slow to a crawl and you risk data loss.
Never go below 5GB and 10GB is better.