What board should I use to control my 3D printer
There doesn't seem to be anything different in your application to any of the 3D extruder-based printer applications. All this has been sorted out many, many times already. 



Most use an Arduino-based board with built-in stepper drivers and temperature control interface. Marlin firmware, for example, allows monitoring of position and temperature on the printer's LCD so that you don't have to leave the laptop connected - although you can if you want to. 







simulate this circuit &ndash; Schematic created using CircuitLab

The slicer software doesn't generate a "slicer file", it generates CNC g-code which can be transferred by SD card to the printer which reads and executes line by line. Most slicer software can drive the printer over USB or Ethernet but that ties up the laptop for the duration of the print.