What does the expression кому обещать эту жизнь без потерь mean? Is the subject 'кому who' even though it is in the dative case? Who can promise this life without care? or is it To who can a promise of this life without care be made?
Out of context this phrase can be parsed both ways indeed.
Russian (as well as Latin and some other languages) has so called "dative of the agent", which conveys a meaning of necessity, moral obligation or fate:
Тебе с ним жить // You'll be spending your life with him.
Мне в магазин // I need to go to the store.
Ей ли меня судить? // Is it up to her to be judging me?
However, if we look at the lyrics of the song this phrase is from:
И куда отправлять эти письма теперь?
Если адрес не знаю, дорогу забыла.
И кому обещать эту жизнь без потерь?
Если я потеряю все то, что любила.
, it becomes clear that within this context the phrase means "to whom shall I promise" and кому is an object here.
I don't think there is room for an ambiguous translation of your sentence:
кому обещать эту жизнь без потерь
the most literal translation would be: "To whom this life without losses (or loosing) can be promised?" You suggested phrase without care does not reflect the meaning in this sentence. Having a worriless life and life without loses are too different things.
This sentence does not have a subject, the indirect object is Кому and direct object is жизнь, in Russian the sentence would be classified as indefinite-personal sentence (Wiki source in Russian).