Can I register if I have US citizenship, but am living abroad and don’t have my own permanent address in the USA?

YES. In this case you must register at the last address you had before you left the USA, or at the address to which you will return. If the latter, you may not simply pick the place you would like to return to in an ideal world; there must be some indication of a present intent to return and live there (e.g. vehicle registration, driver’s license, property ownership, family, etc). If you have never had a permanent address in the United States but are a citizen you are allowed to register at an address you plan on moving to or an address where you currently have family living. The Overseas Vote Foundation provides excellent information and resources on all matters related to registering and voting from abroad.