[RPG] How/when does an ability like a dragon’s breath weapon recharge, if it’s listed as “Recharge 5–6”

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.

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 are a unique ability of dragons that are part of that creature's mix of unique features, as is a Gorgon's breath (et cetera). Per the "Limited Usage" heading on Monster Manual p. 11 (or the corresponding section of the Basic Rules):

Recharge X–Y.

The notation "Recharge X–Y" means a monster can use a special ability once and that the ability then has a random chance of recharging during each subsequent round of combat. 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 finishes a short or long rest.
For example, "Recharge 5–6" means a monster can use the special ability once. Then, at the start of the monster's turn, it regains the use of that ability if it rolls a 5 or 6 on a d6.

The dragon does not have to expend its breath when readied. The Ready action states:

When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger. Remember that you can take only one reaction per round. (Ready, SRD, p. 93)

The text that follows covers spells and those features which makes the spell provision for Ready (burn the spell slot) a case of specific over general. The dragon's breath is not a spell, so it is not covered in that specific exception to the Ready action mechanic.

Since the description in the Monster Manual does not indicate ability loss when not used under a Ready action (temporary loss of that ability is folded into the Recharge mechanic), and since other abilities (like attacks, lair actions, etc.) are not lost by not activating them when Readied, there is no reason to expend the breath when it was readied but not used.

If you view the breath as a special attack, like a wing attack, or a tail attack, this is consistent.