Dragon breath weapons have a recharge time. For example, a very young black dragon can use its breath weapon every 1d4 rounds.
Do you re-roll the recharge time each time you use the breath weapon? Or once per (...encounter? ...day? ...dragon?)
I don't have a rules citation for your second question, but I've always seen it played that you reroll the recharge die after every breath.
As to your first question -- from the dragon listing:
Once a dragon breathes, it can’t breathe again until 1d4 rounds later. If a dragon has more than one type of breath weapon, it still can breathe only once every 1d4 rounds.
If a dragon rolls all ones, we can replace this with:
Once a dragon breathes, it can’t breathe again until one round later. If a dragon has more than one type of breath weapon, it still can breathe only once every round.
so I think the answer is: if you roll a 1 on the 1d4, the dragon can use its breath weapon next round.
Yes, you re-roll the 1d4 each use of your breath weapon
The key to understanding this is in this line from the Breath Weapon special abilities entry: (emphasis my own)
Most creatures with breath weapons are limited to a number of uses per day or by a minimum length of time that must pass between uses.
So that's one roll of your cool down die each and every time you use your breath weapon, unless the specific creature entry says otherwise.