I can't answer your title question or last question, but I can answer the two in the middle ("Was altering the body to fit a persons gender always the norm for transgender people? What about non-binary or transgender people with no gender dysphoria?") (I also assume you are actually asking about transgender people with body dysphoria / dysmorphia, rather than gender dysphoria in general, since having gender dysphoria is a diagnostic requirement for being diagnosed as transgender).

Many of our gender and sex-related concepts and terminology in english is very new.
"Homosexual" was coined in 1865 by Karl-Maria Kertbeny. "Transgender" was coined in 1965. Before the 1960's, transsexual was the term used in English for people we now call transgender, and it was coined in 1949. The term or idea of being genderqueer - not necessarily having any defined gender identity- was only coined in the 80s and began finding greater usage in the 1990's. "Non-binary" only came about in the 2000s.
So, the answer to "was altering the body to fit a persons gender always the norm for transgender people" could almost be seen as a sort of trick question, since surgery and physical transition has been the norm since before the term "transgender" came into usage. but it was not the norm before the 20th century.
In 1919, Magnus Hirschfeld founded the institute for Sexology in Berlin, which is where many of the first documented gender reassignment/ gender confirming surgeries were carried out in late 1920s. One of the first women to get a gender confirming surgery had a history of attempted self-surgery (such as attempting to tourniquette off her penis as a child), and it was things like this that led sexologists to experiment with enabling physical transition as a treatment for people with body + gender dysphoria.
In 1952, Christine Jorgensen became the first person widely publicized as undergoing sex reassignment surgery, which is when it really started to enter the public consciousness.
Regarding people without body dysphoria, physical transition and surgery was considered very risky and dangerous (many early patients who received gender reaffirming surgeries died as a result, such as Lili Elbe, who you may recognize as the subject of the 2010s film The Danish Girl), as well as very permanent, so it was generally reserved for the more "serious" cases. It would have been extremely difficult for a non-gender-conforming person to get gender-related surgery if they were comfortable in their body (and honestly, I'm not sure why they would seek gender surgery in the first place if they were comfortable with their body and were not experiencing body-related dysphoria).

There are many documented cases before the modern era of people who in a modern context could possibly be considered transgender, such as John/Eleanor Rykener, who was born as John but lived as Eleanor on and off for a period of time in the late 1300s. But it is not possible to know how they would have identified themselves according to our modern terminology (for example: would John/Eleanor consider theirself genderqueer, genderfluid, transgender, intersex, etc? We have no way of knowing).

Several cultures have third sexes (Indian, Hawaiian, etc) which transgender people often fall into. So, the history of how transgender people have been perceived will be different in the context of those cultures vs. European cultures, which is the group that I am focusing on in this response.