I've seen ATCs use little slides with flight numbers for tracking arrivals and departures - is there a term for them and are they still used today with modern ATC terminals?
Yes they are called flight progress strips or simply strips. They are still in use, though they tend to be replaced more and more by the electronic strip or e-strip.
They are also used by en-route controllers (not only terminal as you mentioned). From my experience from Athens ACC, PALLAS supports e-strips, but there is also a printer that prints them in some intervals (I don't remember what triggers the print). Note that some time ago, during the procedural control era, those were manually completed by the ATCOs. I guess they are still manually completed in TMAs and ACCs that use procedural control as primary method and lack RADAR service.