i think this should also have the flavour text not just the stats

Cannot make the shortsword attack due to injury. Longsword attacks deal 8 slashing damage. May use melee attack (longsword) twice. The Half-Red Dragon may replace its 2 longsword attacks with the Fire Breath weapon.

How is this different than a Dragonborn?

As I understand it, the dragons, the dragonborn, and the kobold are each separate species. Their exact origins vary by setting. A half-dragon however, is a hybrid between a polymorphed dragon and a humanoid.

Going by the normal rules for the spell Polymorph, that would be tricky since the spell only lasts an hour and I think it only allows changing into "beasts" -- not humanoids. The spell True Polymorph on the other hand does offer a way to allow the dragon to keep their humanoid form indefinitely. Ancient Metallic Dragons can already change shape as an action.

Quote from Finchburn >>

As I understand it, the dragons, the dragonborn, and the kobold are each separate species. Their exact origins vary by setting. A half-dragon however, is a hybrid between a polymorphed dragon and a humanoid.

Going by the normal rules for the spell Polymorph, that would be tricky since the spell only lasts an hour and I think it only allows changing into "beasts" -- not humanoids. The spell True Polymorph on the other hand does offer a way to allow the dragon to keep their humanoid form indefinitely. Ancient Metallic Dragons can already change shape as an action.

Basically.  They can also be created through magic rituals/events, but that’s much rarer.

You can find the template in the “H” part of the “Monsters” section of the Basic Rules.  Lore-wise, it can be applied to any Beast, Giant, Humanoid, or monstrosity.

(Although, practically speaking, if a player wanted to play a half-dragon, I’d probably just give them tweaked Dragonborn stats.)

Half Dragons have tails, Dragonborn do not.

If a Half-dragon is created through rituals and stuff, could a character become one?

Quote from mehi >>

If a Half-dragon is created through rituals and stuff, could a character become one?

 Yes, at GM discretion, a player may acquire this template.

It is a bit overpowered at low levels though.  Definitely a major quest to acquire and set up the ritual

Dragonborn are their own separate race. And they don't have tails

 How is the DC for the breath weapon higher than the DC for the red wyrmling?

Quote from Musitoro >>

 How is the DC for the breath weapon higher than the DC for the red wyrmling?

 Because whichever ability score the DC is tied to (not sure which one it is) is higher on the Veteran NPC stat block than on the Red Wyrmling statblock.

If Half-Dragons are the result of a polymorphed dragon mating with a human and the dragons represented include chromatic ones, it would seem to apply the rules of shape changing used by metallic dragons to chromatic ones. Have the chromatic dragon RAW details been changed to support shape change or am I missing something??

Chromatic dragons don't have shape changing abilities. However, biological mating is not the only way for half dragons to be made. They can come into being as a result of a spell gone wrong or a ritual consisting of bathing the desired dragon type's blood.

Additionally, there will always be exceptional creatures. Some chromatic dragons may have studied the arcane arts, follow tiamat close enough for cleric levels, or have unlocked an inner power sorcerer. No matter the source, it is possible for chromatic to also change their shape. Just not the average chromatic.

Quote from ExplosiveCarnage >>

Chromatic dragons don't have shape changing abilities. However, biological mating is not the only way for half dragons to be made. They can come into being as a result of a spell gone wrong or a ritual consisting of bathing the desired dragon type's blood.

Additionally, there will always be exceptional creatures. Some chromatic dragons may have studied the arcane arts, follow tiamat close enough for cleric levels, or have unlocked an inner power sorcerer. No matter the source, it is possible for chromatic to also change their shape. Just not the average chromatic. 

 On the vein of artificially created half dragon, Bahamut is known to reward exceptional members of his cults with conversion into a half dragons.

(These reborn followers are sometimes confusingly called, “Dragonborn”, although they have no relation to the race with the same name.)

Where's the stat sheet so I can create a custom half dragon character for my mate

Quote from fuzzybron >>

Where's the stat sheet so I can create a custom half dragon character for my mate

 Go to the Basic Rules in the Compendium, and look under “H” in the monster listings.

Quote from Musitoro >>

 How is the DC for the breath weapon higher than the DC for the red wyrmling?

Likely because the DC is a typo.  By the rules for other members of the Dragon family, the DC equals 8 + Prof + Con Mod, which would be 13 in this case, instead of the 15 used in the stat block.

Also DCs used on monster statblocks don't follow rigid rules, they can be higher or lower than what you'd expect them to be. And this is technically an official but still custom creature.

I'm using this as an NPC ally in a campaign, who is a Dragonborn veteran. Then I realized I could use this. Pretty handy. Although I might make his breath weapon do a bit less damage

In that case, you could have it scale like Dragonborn, but maybe an extra d6