Axis & Allies Developer and Playtester; War of the Ring Editor and Playtester
Does that mean you can move the token you’ve literally just placed?
Yes, you may move any or all Ally tokens on the board, including the one you just placed.
Also, to clarify, all tokens previously placed can be moved anywhere on the board?
They may not be moved anywhere. Ally tokens move from their current location to a connected location. See page 14 of the Rulebook.