How to get & train a service dog for anxiety or PTSD? Best breeds?

There are a number of things to consider.

#1- a "service dog" is a dog trained to perform certain tasks to assist the handler/owner, due to a disability (either physical or psychological).
examples;
a person in a wheelchair with a dog who can open doors, push elevator buttons, etc.
a blind person with a dog who helps them cross the street safely
a dog who can sense that a person is about to have a seizure, alerts the person so they can get to a safe location
a dog who senses a panic attack is about to occur and can behave so as to prevent the panic attack from happening

#2 - a "service dog" as described above is different than an "emotional support" dog or "therapy" dog. These dogs provide comfort and emotional support and are not trained to perform a specific task.

See this article which explains this more specifically:
https://adata.org/publication/service-animals-booklet

Regarding your question about riding horses, I am not sure how that would work or what exactly you would be expecting (the dog to walk along next to you and the horse?) I think the biggest concern here would be the safety of the dog and then temperament of the horse (would the dog spook the horse, which puts all 3 of you in danger). My first thought would be this is not advisable, but whether or not it is allowed is probably up to the facility in which you are riding.

Generally Labrador retrievers and golden retrievers make great service dogs, though any breed can be a great service dog.

A service dog is not required to be "registered" per say. See the following:
https://www.ada.gov/service_animals_2010.htm
https://www.ada.gov/regs2010/service_animal_qa.html#cert

If you are wanting to train the dog yourself, check out this book:
https://www.amazon.com/Training-Your-Own-Service-Dog-ebook/dp/B015SJ32AM/ref=as_li_ss_tl?_encoding=UTF8&pd_rd_i=B015SJ32AM&pd_rd_r=ACG5MWKYACEVMPXV9BNT&pd_rd_w=5Umlr&pd_rd_wg=W4y6T&psc=1&refRID=ACG5MWKYACEVMPXV9BNT&linkCode=sl1&tag=rove05-20&linkId=2f897c47d415402a56122c1e082eae13

I hope that helps you. Thank you for asking PetCoach!