Getting the code package right is largely about not over-specifying. ASME Section VIII is essentially universal in North America. TEMA class should match service severity. 3-A and ASME BPE apply to hygienic work and are not interchangeable. API 660, PED and Canadian CRN are jurisdiction or industry driven, and each adds cost and schedule that should be a deliberate choice.
In most North American jurisdictions, pressure vessels above certain size and pressure thresholds must be code stamped and registered, and your insurer will generally require it regardless. Small low-pressure units may fall outside, but the stamp is inexpensive relative to the cost of discovering you needed it.
The Manufacturer's Data Report for a Section VIII Division 1 pressure vessel. It records design conditions, materials, construction details, the inspector's certification and the National Board number. It is the exchanger's birth certificate and it is what you need when specifying a replacement bundle years later.
Yes, and many do. They govern different aspects: pressure integrity, mechanical construction, and hygienic design respectively. Meeting all three means satisfying all three sets of requirements, which is normal on sanitary process equipment.