[RPG] Can one use a breath weapon and a melee attack in the same turn

It’s practically impossible (-20 penalty) to hide while attacking, running or charging.

So you can remain hidden while making a spring attack, but the DC to spot you is not going to be very high. If you still manage to win the opposed roll, your target will be denied its Dexterity bonus to AC and will have an additional -2 penalty because you are effectively invisible.

From Skip Williams' column:

If a foe you cannot see hits you with a melee attack and is adjacent to you at the time, you know the foe's location. For this reason, smart foes move right after they attack; even a foe that has made a full attack can move after attacking by taking 5-foot step (provided it has not already moved during its turn).

When an unseen foe hits you with a melee attack from more than 5 feet away, you know the general direction from which the attack came and that the attack came from more than 5 feet away, but you do not know the attacker's location.

That is the main reason to spring attack: you need to move after attacking, even if invisible, otherwise your target will know your location. If your target has failed its spot check, you still have total concealment (50% miss chance).

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.