For me, it's yes and no. Yes on internal hybrids but no on external hybrids. External hybrids are childern born from a mix-race couple that possess physical attributes from both parents. Why I say no to external hybrids is because on the show one of the rule they follow is no human allow in the ponyverse of MLP:FIM. Which mean that although the Minotaurs(human + bovine) were a hybrid race in greek mythology, they would be their own race in MLP:FIM. On the other hand, Internal hybrids are children that generally possess physical attribute from one of their parent. Even though they look normal, if you test their dna you would see that they are in fact a hybrid. For example is the cake twin for even though they were born from earth pony parents, one was born a pegasus(Pound Cake) while the other was a unicorn(Pumpkin Cake). If external hybrids were possible in MLP:FIM then the cake twin would have been born as alicorns. Another example is the Pie family, they were shown to eat rock soup and Maud Pie was shown eating a diamond. So even though they are earth pony, they possess the eating habit of dragons. So the Pie family could have a draconic ancestor.

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If external hybrids were possible in MLP:FIM then the cake twin would have been born as alicorns.

Not necessarily. If alicorns can be produced through genetics then it is entirely possible for matings of ponies to never result in alicorns. The reason for this comes down to epigenetics, or multiple genes coding for expression of phenotypic traits. If the genes found in all ponies that encode for their phenotypic expression of race are, say, linked to genes A, B, C, and D, (were A is dominant to B, which is dominant to C, and D is dominant to all) and receiving a D instead of a d results in the phenotypic expression of being an alicorn, then it would literally be imposable for any paring of ponies that did not involve an alicorn to result in an alicorn as all non-alicorns would be dd.

But to answer the question fully, I will present a message I sent to a friend about this topic:

I full well know that there is no force in heaven nor hell that could cause a dragon and a pony to reproduce due to genetic differences. It would require both species to have; 1) the same number of chromosomes, 2) chromosomal composition that is nontoxic to paired chromosomes, 3) similar coding mechanisms and language for coding, and finally, 4) sperm implantation into an egg due to homologous signaling between the egg, sperm, and uterus.

But I would like to say, if you want to make hybrids, use whatever rules you want, this is fiction, anything is possible. Just bear in mind, if you want full, 100% realism, hybrids are literally imposable unless they are functionally the same species in terms of chromosomal, composition, count, and coding, and poses the same signaling responsible for implantation of sperm into an egg.

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Guess would is a little strong here. What I mean is that if external hybrids exist then since the cake twin possess genes from all three pony tribes which create an option of them being born alicorns. I believe Flurry Heart is an internal hybrid since her father is a unicorn and her mother an alicorn. So Flurry Heart was born an alicorn because she take after her mother.

For the second part I have two theories. First one would be magic can generally get around the laws of biology. So what is not possible in our magical-less world could be possible in a magical one. The second one would be even though some race look to be different, they could be from the same species. Like the vampires, werewolves, and human in the Underworld franchise. Even though they're different, they're in fact from the same species(Homo Sapien Sapien).

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As I said, go by whatever rule set for reproduction you want to use. It is fiction after all, so there is literally nothing that can be claimed as posable or imposable definitively. I was simply trying to point out that as you increase realism, certain things need to get taken into consideration.
{I also just now took into consideration the "For me" in the original post as the first words}

Well we know mules and zebriods are for the simple fact they are real, it also safe to say any equine hybrids are possible ,like half-kirins and half-breezies, and I would even go as far as saying half hippogriff for both equines and for griffons. The comics say that half-dragons are real and sense it is the default in fantasy settings for dragons to able to breed with anything I would say it would be stranger if dragons couldn't breed with anything in MLP. Being shape shifters changelings can probably breed with anything as well, for the any other half-breeds they would probably need magic to make them and they probably do have magic items, spells, and potions to help with that. In fact we seen a few spells in the show that could probably do the job.

Also I'm going to say it because know it is true, Discord can have kids with anything.

I would dispute that "No External hybrids exist" because of three examples they are as follows"

• Hypogriffs,
• kirins, and
• Alicorns.

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Just like with the minotaurs, the hippogriff could be their own species and not a crossbreed between pony and griffon. And as speciesist as Chancellor Neighsayer was, he never refered to the hippogriff as a degenerated mudblood. From what I seen in Sound of Silence, the show seem to implied that the kirins are not a crossbreed between pony and dragon but a subspecies of pony, likely a unicorn subspecies. Alicorns seem to be a natural/magical mutation with Candance(pegasus) and Twilight(unicorn) evoluting from their pony tribe into alicorns. Or alicorns could be the progenitor of the three pony tribe and their subspecies; unicorn/kirin, pegasus/threstal, and earth pony/crystal pony.

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I'd say Gallus is a little bit external. No other griffon has their cat half a color that is unnatural for a cat to have, and no other griffon except Gabby is the same color on their whole body. So he's probably part pony and Gabby might be too, her hoping for a cutie mark would make more sense this way.

Treating the three kinds of pony as different races is a non-starter. Pegasi have a different number of limbs, if it was a matter of genetic difference they wouldn't be remotely the same species. No, with the Cake twins we have it firmly established that it's something akin to sex or bee caste - a matter of a genetic switch activating or not, or womb conditions.

Alicorns of course are independent of genetics. Otherwise you couldn't ascent. Probably exposure to extreme magic levels plays a role, but could be something like being straight up divinely chosen.

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Hippogriffs and kirins aren't at all suggested to be in any way hybrids. They are their own species, though related ones.