What kind of "support" does this refer to? Physical, financial, emotional? Or is the meaning as ambiguous as it is in English?
Happy "1 year unbroken Lingo-ing!"
Could mannen be the subject of this sentence as well? (Thus does the word order dictate that kvinnen is the subject?)
Yes, "kvinnen" is the subject here, and this sentence as it is cannot have "mannen" as a subject. :)
"Vi burde alle støtte hverandre": is this the correct usage of "should"?
Yes, both "burde" and "bør" would work in that sentence.