There is no rule which forces you to use a 3 hex tile, you can indeed fill such a spot with three 1 hex tiles if you like. The exception to refill the board is probably for the case as described in the example below it, or a 2 worker spot with only 3 hex tiles left.
Correct. You only refill the tile board during phase II if a situation arises in which you still have workers remaining in the field but cannot place an available tile (e.g. single worker and only 2 or 3 hex tiles remaining, two workers and only 3 hex tiles remaining).
You do refill the board each round as part of setup though. That is kind of easy to overlook as the rules for doing that are separate from the rules for phase II.