Just phonetics. You often hear -ю forms in poetry and songs (where they also give the poet freedom to add a syllable). Now they are mostly seen as poetical langauge. You sometimes read them in books, especially older ones. And sometimes people use them in speech - though it depends on rhythm and the word in question (usually it is nouns: судьбою, сам с собою, etc.). Where they are unusual they can be used for fun - but in some words some can use them without noticing. I think before the Revolution -ю was also supported by that people heard it in church. Church used Slavonic that preserved -ю forms. Slavonic was an important literary langauge till 18th century and the sourse of many forms in literary Russian and in poetry. In this case it is not exactly 'source', rather one more reason to preserve it.

In russian language we usually use "своей, моей, твоей". Words "своею, моею, твоею" are archaisms or something like that, but it won't be a mistake if you use it :)

I wouldn't say it is archaic, it is used in everyday language with these pronouns as a kind of lazy stretching the last syllable to ease connection of words. But with adjectives ("красивою девушкой") it is an archaism, used in poetry, such a folk style.

Can we use it with ваш/наш.. этот and other words as well (вашею/этою?) Is it as archaic as like English "thou/thine"? - that is only used in old movies and books... or is it like "conversate" - something people usually say but is non official?

In russian language we usually use "своей, моей, твоей". Words "своею, моею, твоею" are archaisms or something like that, but it won't be a mistake if you use it :) If you want to learn little bit Russian with a native speaker feel free to hit me up! I am not a teacher, but it could be an interesting exp.