If I were to cast the Dragon Breath spell one another character, if it were set to the element of my Draconic sorcerer, would it benefit from the damage boost that Dragon Sorcerers get at 6th level? For that matter, would it benefit from the Elemental Adept feat?

I'd say yes, because Dragon's Breath is still your spell.  You cast it, you're holding the concentration... the other player just activates it.

Draconic Sorcery's Elemental Affinity states:

Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.

Elemental Adept:

• Spells you cast ignore resistance to damage of the chosen type. In addition, when you roll damage for a spell you cast that deals damage of that type, you can treat any 1 on a damage die as a 2.

I would argue "no". Dragon's Breath does not benefit from either of these.

Look carefully at the wording. Elemental Affinity states that *spells* you cast have your Charisma modifier added to the damage roll. Elemental Adept also says that *spells* you cast ignore resistance to damage of the chosen type. Dragon's Breath grants an action to essentially allow someone to repeatedly use an ability like a dragonborn's breath weapon; the spell itself does not deal damage. The spell is simply granting the ability to use that action, and the action is what causes the damage.

Quote from karterfone >>

I would argue "no". Dragon's Breath does not benefit from either of these.

Look carefully at the wording. Elemental Affinity states that *spells* you cast have your Charisma modifier added to the damage roll. Elemental Adept also says that *spells* you cast ignore resistance to damage of the chosen type. Dragon's Breath grants an action to essentially allow someone to repeatedly use an ability like a dragonborn's breath weapon; the spell itself does not deal damage. The spell is simply granting the ability to use that action, and the action is what causes the damage.

It's worth thinking about RAI. It doesn't make sense to create a spell that gives a dragon sorcerer dragon's breath and they don't get to add Elemental Affinity. A concentration spell, once cast, doesn't stop being a spell. You could cast dispel magic on a character with dragon's breath or crown of stars or any other number of spells.

Quote from Matthias_von_Schwarzwald >>

Quote from karterfone >>

I would argue "no". Dragon's Breath does not benefit from either of these.

Look carefully at the wording. Elemental Affinity states that *spells* you cast have your Charisma modifier added to the damage roll. Elemental Adept also says that *spells* you cast ignore resistance to damage of the chosen type. Dragon's Breath grants an action to essentially allow someone to repeatedly use an ability like a dragonborn's breath weapon; the spell itself does not deal damage. The spell is simply granting the ability to use that action, and the action is what causes the damage.

 So, Melf's minute meteors wouldn't get Elemental Affinity or Elemental Adept bonuses either?  And goodberry and regenerate's ongoing effect wouldn't benefit from Disciple of Life? 

Could I cast dispel magic on Melf's minute meteors?

As I see it: no, no, and yes.

Sooo... when a spell does something, and you can use that something to do dmg, it doesnt get boosted when something says "spells that deal damage";
but when you use a spell that makes a fruit (something), and the fruit (that something)heals, it gets boosted when something says "spell that heal"
Spell --> thing--> effect. Effect may or may not get boosted. Awesome

Dragon Breath is the spell that grants the action. IMO this is silly of course it does. Unless I see Sage Advice say otherwise I'm giving Players their abilities because as a DM I'm using the same ruling. Taking away abilities unless it's a crystal clear game breaking just creates a bad gaming experience.

Quote from Astromancer >>

Could I cast dispel magic on Melf's minute meteors?

Yes, Counterspell would be better but yes.

That was the point I was trying to make to karterfone. As it Melf's minute meteors is a spell, even on subsequent rounds you would be able to. Therefore, you could apply Elemental Affinity or Elemental Adept to it. This logic would apply to other spells like dragon's breath, which are both spells that give actions.

FMB stated it clearly. 

If a spell that grants an action /remains in play / is permanent, It is still something the caster is responsible for, like the DC. Yes even with dragonsbreath. Its their spell and if they have a way to buff it as Thucy asks and FMB answered then buffed it will be. As would any ongoing free standing or seconded spells because the caster hasnt altered despite the spells location being mobile / long out of sight / on another plane.

This has been corroborated with the horrifying 'clarification' on goodberry benefiting from Disciple of Life and healing 4 hitpoints per berry. Its a free standing spell that others consume to activate with an action and apparently gain 4 hitpoints instead of one from and thats with multiclass abilities not a draconic element spell cast by draconic sorcerer.

Just remember, only one damage die gets the boost, I'd probably rule the first breath gets your Cha damage but not any use after

I am curious about this spell called dragon's breath I run a dragon sorcerer

And it specifically States that if I touch one willing creature

It never states that I could not cast it on myself by touching myself

It never says that you cannot self cast

Any insight on that question

You can cast it on yourself or on another willing creature.  In my experience, Dragon Breath is far more effective when it's cast on an ally instead of the spell caster using it on himself though.  I love that spell, I've only cast it on myself once, and I never will again unless it's an emergency.

Any time you cast a beneficial spell with a range of something greater than self, you may still select yourself unless the spell description specifically states otherwise.