This is explained under "Limited Usage" on page 11 of the Monster Manual or here in the basic rules on D&D Beyond:
Recharge X–Y. [...] At the start of each of the monster's turns, roll a d6. If the roll is one of the numbers in the recharge notation, the monster regains the use of the special ability. The ability also recharges when the monster ...
Dragon breath isn't a spell
Dragons can cast spells if the DM activates that optional feature in the MM; by default a dragon's breath is not a spell attack. Trying to apply the spell rules linked to the Ready action to an action that isn't a spell can be a table rule or a house rule - you could do it that way, but the rules don't call for it.
Breath weapons ...
The dragon cannot use the second breath weapon while the first is on cooldown. From the breath weapon rules:
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.
Only after it is triggered.
When you do a Ready action, you choose a trigger and an action (Attack, Ability, Cast a Spell, etc). However, you might not use that action at all, and so don't use its resources. For Recharge X-Y abilities, you only recharge them after you use them.
First, you decide what perceivable circumstance will trigger your reaction. Then,...
The rules are unfortunately unclear here
Possess Corpse is listed as an action for the dybbuk and actions are a part of a monster's statistics (see this Q&A for more on what counts as a statistic):
When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures....
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 ...