Because "Tochter" is a feminine noun.
"Danken" is one of the few verbs after which you have to use the dative case (and not the accusative) for the direct object.
Masculine/neuter nouns: dem
Feminine nouns: der
Plural: den (+ the noun itself adds an "-(e)n" if its plural does not end in -s or -n already, i.e. "Ich danke den MännerN")
See also this chart:
Yes! Quite a few actually, you can find more here:
"Danken" (to thank) is a so-called dative verb, i.e. it belongs to a special group of verbs that trigger the dative case. "Tochter" (daughter) is a feminine noun, and the definite article ("the") for dative feminine is "der". See this table:
As you can see from the table in the link, some of the articles are used for several different things. "Der", for instance, is used for masculine nouns in the nominative case, but it's also used for feminine and plural nouns in some of the other cases. So "der" doesn't always refer to masculine nouns.
For more information on dative verbs, see: http://german.about.com/library/verbs/blverb_dativ.htm
There's no strict rule which case is to be used. But at least you can feel that something is given to someone. Giving objects to someone implies the dative, even if it's something immaterial like "a thank" (which, unlike English, is an object in German).
No, it's not. It depends on the context whose daughter she actually is. In this case the family background seems to be clear (not to me, though). Perhaps it goes like this: "Ich danke der Tochter, doch ich küsse den Sohn".