The Manufacturer of a device to be marketed in the US must have registered as an Establishment with the FDA. That's the HQ location, not the US office. That's done online. There's a fee per calendar year, payable starting in October.
They also must List their products, just as any other Manufacturer must. That's also done online. This requires that the Manufacturer know the products' FDA class and product code. If any of the products are Class II or III, regulatory clearances/approvals must be in place already. For a Class II product, that's normally a 510(k). For a Class III product, that's normally a PMA.
Every Establishment located outside the US must have a US Agent, who serves as the official point of contact if for some reason the FDA is not able to reach the foreign HQ location in a timely manner. Agents basically do nothing 99.9% of the time. If the Manufacturer does not have a US office, the Agent normally is either arranged via a contract relationship, or is the US initial importer/master distributor. If the Manufacturer has a US office, that office should be listed as the Agent.
All FDA Manufacturers must conform to 21 CFR 820 and all other applicable US regulations, no matter where they are located. Manufacturing must be conformant to the GMPs. If one or more contract manufacturers are used, those manufacturers must be effectively controlled and must operate in conformance with the GMPs. Labeling must comply with FDA regulations, which are significantly different from MDD requirements. The FDA may inspect foreign manufacturers. In theory this can be done with no advance notice.
In most cases, a well-run MDD/13485-conformant design and manufacturing control system, plus product documentation in the form of a change controlled Tech File or Design Dossier, plus retention of all design and manufacturing records, will be adequate as the required QS plus DMR, DHF and DHR.
In most cases, a well-run MDD-13485-conformant adverse event reporting system will be easily adapted to meet FDA requirements, which are conceptually similar but which differ in a number of details and requirements.