You don't get Puerto Rican senate representation if you move there but you can live in Puerto Rico and still vote for president and the Senators for the State you are registered in.
No, you cannot.- If you legally reside at a home you still rent in Iowa, you can vote with that as your legal address, whether you are currently staying in San Juan, Atlanta, Tokyo, or Paris.
- If you end that lease in Iowa and you move to Iceland, you can still absentee vote as a US citizen and former Iowa resident, using the address of the Iowa home you no longer rent.
- If you end that lease in Iowa and move somewhere else in the US, be it Honolulu, Puerto Rico, Wyoming, or DC, you must register as a voter in your new US jurisdiction, where you are entitled to vote for the seats allotted to your current area of residence.
If you live within the United States (and Puerto Rico is within the United States), it is voter fraud to register and vote using an address that is not yours (be it rent, own, whatever). So if you MOVE to Puerto Rico, you must register to vote as a resident of Puerto Rico, like every other US citizen that lives there.
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That's no difference than moving to Iceland.
Iceland is not part of the United States! That's the difference!