All the Hydra are affected, but it can only activate the spell once per turn and can only make one cone at a time.
A Hydra (or any monster) has only one action per turn and nothing in its block of statistics gives it multiple actions. Create a cone using The breath of the dragon is his own action because the spell asks a creature to "use an action to exhale energy." Thus, the hydra can only activate The breath of the dragon once in his turn. The multitude of heads will not help the hydra here; having them all attacked on the same turn forces the Hydra to take the specific "Multiattack" action, which it does not do here.
Only one damaging cone can be made at a time
the The breath of the dragon spell states:
You touch a consenting creature and give it the power to spit magical energy. of his stuffy, provided that he has one. Choose acid, cold, fire, flash or poison. Until the spell ends, the creature can use an action to exhale energy of the chosen type. in a 15 foot cone. Each creature in this area must make a Dexterity save roll, taking 3d6 damage of the chosen type in case of backup failure, or half less damage if successful.
The creation of a single cone is probably intended for the singular used in the description of the spell: "his stuffy","a cone ","this zone".
Apart from the possible intent, the spell indicates that the creature creates "a cone (singular)" and not "a cone for each head of the creature" and thus creates a (single) cone even though we read this rule strictly.
And so, as stated above, Hydra can use its action to activate this spell once at each of its turns and this activation creates a single cone.
To answer your other question ("Can each head have a different type of damage?"), When the spell is cast, the caster selects a unique type of damage and the creature exhales energy "of the chosen type" (and inflicts damage "of the chosen type"). Thus, the creature uses the type of damage chosen by the caster and, as its heads are part of the creature, each head also uses the same type of damage.
This is probably the case that all heads can make this cone damageable. The spell affects the entire creature and allows it to spit energy from its mouth, it does not specify that the same mouth should do it each time (although I doubt that the developers have thought of this situation). In addition, if you hear the phrase "A Hydra can spit energy from her mouth", you will assume that it means any of her mouths.
If instead you interpret "his mouth" as strictly singular, you encounter a problem because a Hydra has nothing that can be described as "his mouth", but only "one of his mouths".
The spell clearly assumes that an affected creature has only one mouth and, therefore, which mouth (s) of a multi-mouthed creature can activate the spell that should be left to your GM.