Do captains and first officers always fly together or are they mixed?
In airlines where pilots bid on monthly lines it is conceivable that a pair of pilots fly all month together. However, trip trading and reassignments rarely let this happen. It is far more common that a crew only stays together for a single "pairing" that lasts between 1 and 4 days (domestic 121 USA). 

It's not a drawback to fly with new people because training emphasizes standard flying according to the FOM. Ideally, every pilot is exactly the same.
  Practically it is close enough that you can drop any two pilots from the same airline into an airplane and they more or less operate it exactly the same way.