"Her father is a farmer."
Translation:Hendes far er landmand.
Both are considered correct here, but generally you don't need to say the "en" when you're talking about professions in this context. Like you could also say "he is farmer" instead of "he is a farmer".
You can't say "he is farmer" in English; the indefinite article must always be included before the profession: "he is a farmer" (or definite: "he is the farmer")
"sine" would definitely not work here, although I'm not sure I can fully explain why :-)
Maybe some examples where "sine" (or "sin") could be used will allow someone to draw some conclusions:
"Pigen tog sine ting" = "The girl took her thing "Hun forlod sine unger" = "She left her kids"
One thing is that "sine" is a plural form, where as "sin" would be singular:
"Hun elsker sin far, som er landmand" = "She loves her dad, who is a farmer"
Seems to me that "sin/sine" has to have something else in the sentence to latch on to, e.g. Hun or Pigen above, in contrast to the use of Mig or Min/Mine (Me, My/Mine)?
"Min far er landmand" = "My dad is a farmer" would work just fine.
Danish is indeed a complicated language :-)
Sine is a "reflexive possessive pronoun", which means it points back to another pronoun. If there's no other pronoun for it to point at, then you can't use it. Wiki has an article on reflexive pronouns, and if you read that, and follow the link to danish examples, it will probably stick. [Edit. ANother noun. It doesn't have to point at a pronoun, just at a noun.]