Do I need to run the FSDreamteam Update any time I install a new airport?
You don't have to *perform* an actual GSX update (if there's no need for it) but yes, you must at least start the FSDT Installer, and run the "Exclude 3rd party" routine, to be sure the new airport will be Disabled from GSX jetway replacement, and if it's not conforming to the official naming standard (so it wasn't recognized automatically), add its code manually in that page and Apply the changes.
Yes, performing an Update will do that as well but, it will be slower (if you don't need to update GSX), and it will miss airports with non-confirming names (usually freeware), which must always be added manually.