Is there something in Mandarin like "I don't give a f***", something that expresses the same sense of "I don't care" but with more force.
No, but in the dialect spoken where I lived people say 没得屌事.
Strong but not vulgar answer: 我才不管
I don't give a f*** what he thinks
A direct translation would be:"我不在乎".
If you want to make an emphasis or show your anger, you could say:"爱咋咋的". "爱咋咋的" is more commonly used in Northern area, which literally means "[do] anything you like to[, I don't care.]", you can also simply say "随便”.
管他妈的, why didn't anyone suggest this?
Another - not very vulgar - option is 我不管.