There's some protocol that allows remote configuration of a modem from a PC or router. Also, the external MAC (=network) address is usually assigned to the router, not the modem, and some ISPs might be able to simply move your contract including your modem. Depending on how the ISP works, a change in IP address might be necessary however. I'm using DHCP, so I don't know. IIRC, dynamic DNS is usually used inside a LAN to assign names to dynamically generated IP addresses from the DHCP server in the NAT. BTW, every network device has it's own MAC address, because the Ethernet protocol's packet frame uses only MAC addresses, not IP addresses. IP addresses are only used by the IP layer, which sits on top of the Ethernet layer. HTH
p.s.: I don't know how the home network infrastructure looks in the US, I can speak only for Germany. For instance, there might be services that don't use NAT at all (some guy on here said that about ComCast or some other US service). In that case, every device must get its own IP address from the ISP. In that case, the router can be pretty dumb, b/c it only has to forward Ethernet frames. The ISP's DHCP service is used in that case if addresses are to be assigned automatically. In that case, Dynamic DNS remembers names for your local machines relative to some domain name. The modem can be pretty dumb then too, since all it does is muxing/demuxing the DSL on the phone (or cable) line. Even authentication at the ISP need not be necessary if the phone line is entirely known and/or managed by the ISP. In that case, moving would involve reprogramming or exchanging the modem.