1) I would say it's probably most like traditional courting, though there is a certain similarity to say... the Dalish elves in that one would have to be proven to be a contributing member of society to be a desirable spouse. In the case of Imenry's parents her father pursued her mother who was at first disinterested in being tied down. :P
2) Her sword most definitely as she would never be without it. After that it would be necessary items like a blanket or skinning tools so she could make her own blanket.
3) Given the example she had growing up she would probably be a good mother. She's very caring in her way, though she has a rather blunt way of phrasing things so she would probably scar her children with the "sex talk". Still, she would not spoil them (much), and would be strict about some things. She would most definitely teach them self defense and she would probably never bake cookies XD
4) Mostly she just wants someone who will love and accept her, something that she hasn't had since her parents die. Though she also needs someone who can match her in strength and ability, who can meet her stubborn personality and push past it, someone who makes her feel passionately.
5) She wants to know that someone cares about her, since for the past few years all her relationships have been momentary. But she'd also like to be understood, rather than judged.