I would like to know if my analysis and interpretation is accurate or not.
この this → 人(person)の(possessive marker)家族(family)『は』(topic marker):
This person’s family is the topic of the sentence, what the sentence is about.
一体(what the hell)どんな(what kind of)気持ち(feeling)『で』(manner of action):
How the heck do they feel (the family of the person)
100万円(100x10,000 yen) 出した(to take out) んだ (casual of のだ, used for emphasis):
to have paid 100万円.
How the hell does that person’s family feel about having paid 1000000 yen?
P.S.: Is there a difference between 気持ち and 気持? I have two separate entries in my dictionary but as far from I can tell they are identical.