1) They yapped the day away sounds a little odd, however, they gossiped the day away doesn't. Yapped isn't really a word often used, perhaps use chatted.
"cry the sadness away" sounds very poetic, if that was your intention :)
2) My bill is 60 dollars is the correct way of saying it. You can say "what does my bill come to?" or "how much is the bill?"
3) Can I have 40 charged to my card sounds perfectly fine :) I usually say "Can I pay part cash, part card?" and if they say yes, I'd say "Ok, may I pay 20 by cash and the rest on card please?"