Is it true that cockpit doors are bulletproof?
For all air transport operating under 14 CFR 121, 121.313(f) requires a door to be installed between passenger compartment and cockpit.
14 CFR 25.795(a) outlines the relevant design requirements for such a door:

(a) Protection of flightcrew compartment. If a flightdeck door is required by operating rules:
(1) The bulkhead, door, and any other accessible boundary separating the flightcrew compartment from occupied areas must be designed to resist forcible intrusion by unauthorized persons and be capable of withstanding impacts of 300 joules (221.3 foot pounds).
(2) The bulkhead, door, and any other accessible boundary separating the flightcrew compartment from occupied areas must be designed to resist a constant 250 pound (1,113 Newtons) tensile load on accessible handholds, including the doorknob or handle.
(3) The bulkhead, door, and any other boundary separating the flightcrew compartment from any occupied areas must be designed to resist penetration by small arms fire and fragmentation devices to a level equivalent to level IIIa of the National Institute of Justice (NIJ) Standard 0101.04.

So to summarize, any airplane operating under 121 rules (i.e. scheduled air carrier operations) must have bullet-resistant cockpit door.