How do I pet

Always let them sniff you before touching them. Ask the trainer for permission as well, they will usually give it and could also warn you if their pokemon's a biter. It will also give you a legal grounds to sue if you are bit, because the trainer consented yet you were attacked. Then once the pokemon is up for adoption you can discreetly apply for ownership and successfully legally steal their friend.

how does the sylveon have bad mental health?

Because it doesn't like its gender, and so it refuses to cooperate or be reasonable in any way.

They tend to have mentally ill trainers. Eevees do not have the ability to read emotions like a Sylveon does, and eevee only evolve into Sylveon when they wholly trust and cherish their partner.
So once an Eevee evolves into a Sylveon and attains empathic powers, they suddenly sense that the person they trusted and loved the most in life has been manipulating them, and is a disgusting degenerate tranny. Their entire world is shattered in an instant, and they tend to develop extreme trust issues, even with their empathic powers. This is why Sylveon are so rare, most end up ACK-ing themselves like their trainers would inevitably, while the rest are so depressed and detached from reality that no one wants to adopt them.

what about this, same protocol?

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Autism

Pokemon abuse is a serious topic, user, and Eevees are one of humanity's best friends. Sylveon abuse is a tragic tale oft repeated.

what about espeons do they have the same issues

Correct, user. Sylveon trainers have serious levels of autism. They do terrible things, like feed their Eevees strange hormones that make them evolve into Sylveons instead of good, normal Pokemon like Arceus intended.

do umbreon trainers do terrible things

Of course not. This is a problem exclusive to Sylveon trainers.

are there any sylveon trainers that aren't like this?

Yes, but there's a catch. As a psychic and a precog, they're better able to let you know whether they want to be touched or not without resorting to physical harm. So long as you approach without any malicious intent and are respectful to their trainer, it may let you.
The catch however, is that their fur is extremely sensitive, it can even pick up fine movements in the air. This leads to most Espeon refusing to be touched by strangers, so expect to be refused quickly. And especially don't approach them with the plan I laid out earlier, some professors believe Espeon are capable of reading thoughts.

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You could be the very first. The good news is there are plenty of Sylveons to adopt because their owners tend to eat shotgun blasts once their natural testosterone starts overpowering the hormones and they can no longer hide what they really are.

how do I do rehab for them?

what happens if I don't do what's required to pet a sylveon

It's simple, user. You sit them down, and you show them a mirror. And you say, "Like it or not, Sylveon, this is you. And that? That down there? That's your penis. Wearing panties and changing your pronouns doesn't change what you are. It doesn't change how the world perceives you. Instead of asking the entire world to pretend you're something you're not, why not simply accept yourself as you are, and learn to be appreciate of all that you have in life?"

Make sure that Sylveon is nowhere near the internet when you do this, though; if it can access the internet, it will get affirmation from other Transylveons online, and then it will post your personal details so they can harass and possibly murder you. This will obviously impede growth for the Sylveon, and make it impossible for the Pokemon to ever really love itself which can tragically end in death.

he rams his rocket up your ass.

rent-free

rehearsed with an espeon and all he did is walk through the mirror
where do I adopt a sylveon?