Would - på soffan - also work
Sure. In the case of a couch, there's not much of a difference, really.
Is kudden the cushion or the pillow or is there only one word in Swedish
Same for both. :)
Could you say laid the pillow down? this seems like laid is more etymologically similar to lagt, but i get that iy means the same :)
"laid" is fine, though we don't accept "laid down".
Would "She has put HER pillow on the couch" be translated as: "Hon har lagt SIN kudden i soffan"?
Close! It's sin kudde, since possessive pronouns use the indefinite of the nouns they possess. Just like in English: "her the pillow" makes no sense. But otherwise, that's spot on. :)