I have Pokemon Red and I'm playing through it, but I plan to use the Master Balls (both from Red and Gold) on two of the three legendary dogs. Is this possible with just the two games or with both Pokemon Stadiums?
This isn't going to work at all, as there is no way that I'm aware of to trade/transfer items in Gen 1 games, so your master ball from the Red run is stuck there. While I have heard there is some sort of item storage in Stadium 2, my understanding is that it is also tied to your trainer ID, meaning items stored in this way cannot be withdrawn by any savegame that did not deposit the item.
Unfortunately, it is not possible to trade items up to Gen 2 from generation 1. The ability for Pokemon to hold items was not introduced until gen 2, so there is no way to give an item to a Pokemon in gen 1 and then trade it up (however, some Pokemon that you caught in gen 1 and trade up to gen 2 will be carrying items by default, such as a certain berry).
Also, if you have an item held by a Pokemon in gen 2 and then trade it down to gen 1, I believe the item will be lost (even if you trade the pokemon back up to gen 2).