Proper usage of trans-theory terms [closed]

No, but that's already been answered here. To summarize: sex is biological, gender is cultural.

Most people have the same sex and gender, which is why a distinction isn't always made.

Gender is something you have; gender roles are behaviors. Oxford Dictionaries defines gender role as:

The role or behaviour learned by a person as appropriate to their gender, determined by the prevailing cultural norms:
'women's traditional gender roles translated easily into caring for the sick, and nursing became a female profession'

Gender role conflict often happens to people who are not transgender (or, in other terms: cis).

For example, female rugby players are women in a traditionally masculine role (rugby).

You shouldn't be using Urban Dictionary unless you're looking for slang definitions. Use a real dictionary, like Oxford, which defines androgyne as:

an androgynous individual.

• a hermaphrodite.

It should be clear, from that definition, that the difference between an "androgyne" and "gender–non-binary trans-folk" is the same difference between sex and gender.

However, it's important to note what Nonbinary.org says about the term:

Intersex is a physical sex, and androgyne can mean either that, or a gender identity.

Under this definition, androgynes are one type of a nonbinary transgender person. (It's not very descriptive, since different people use it to mean different things.)