[RPG] Can the Dragonborn gain more breath weapon uses per short rest

It should be fine, as long as time allows.

Nothing in the rules says that a long rest cannot immediately follow a short rest. In this case, what the characters are doing during those rests are very different.

Indeed, a short rest and a long rest are what many real-life people do before bed: an hour or so of non-strenuous activity, like reading, and then going to sleep. It's not a stretch to say that a D&D character can't spend an hour studying their magical item, and then go to sleep.

The only time where that might be a problem is if the characters don't have the full 9 hours.

Different things are happening during the two rests.

What's really key here is that the activities during the two rests are different. In the short rest portion, the character is studying the magic item, meditating on it, or whatever is required for attunement. In the long rest portion, the character is doing something else, such as sleeping. The attunement process is still active work, just not active relative to adventuring.

I think that this distinction is why attunement is limited to short rests, and why a character should be able to do chain them together.

No, Extra Attack can only be used to attack. Extra Attack says:

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

Furthermore, the Dragonborn's Breath Weapon ability says:

You can use your action to exhale destructive energy.

If Breath Weapon gave you, for example, a special ranged weapon you could use to make attacks with, this would work. But what it gives you is a special action. You can use that action to use your Breath Weapon, or you can use the Attack action. There's no way to do both with a single action.