Will a dentist always prescribe some type of antibiotic/pain medication after a root canal was performed?

Maybe: Msot of the time the dentist will prescribe a pain medication. Sometimes the dentist will also prescribe an antibiotic. The dentist will decide if any are needed.

Not always: Unless there is a need for antibiotics, they are not routinely prescribed. Pain management can usually be accomplished with over the counter medications.

Not always : Antibiotics are prescribed only if the dentist suspects an active infection. Painkillers on the other hand are mostly prescribed at least for as needed basis.