Yes it is true "hadi" is different from "hadi canım". "Hadi" is use for come on, hurry up !! if someone says something that is exaggerated for you or you can't find believable something that is said you can use "hadi canım". it does not mean "ı don believe you , you are a liar" ; it is not give the lie to anyone. İt means "don't making fun of me".You can use "hadi ordan","hadi lan","atıyosun","sallama" instead of "hadi canım" but these are slang :))) you sholdn't use these slang (my advise)

It depends on the context. So it's important HOW you say it actually. If it is takes someone too long to do something, you can say "hadi canım, hadi!" to tell them to hurry. If someone says something unbelievable, or makes a joke you can't believe, you can say "hadi canım!" to show that you don't believe what that person says. It's like, "come on, don't try to fool me" If you're very surprised after you heard something, again you can say "hadi canım!"

Even though 'hadi' means 'come on' and 'canım' means 'dear', when you say them together "Hadi canım!" with a shocked face, you mean "Oh my god, really?" Imagine that somebody tells you about something, you get surprised and don't believe it, so you say "Hadi canım!". It's like "Oh my god! Really? I can't believe.".

yeap u can say 'hadi' to want to say come on and start walking. but otherway u can use 'hadi canım' different means for instance when u are surprised u can say 'hadi canım!' or as u said when someone hesitates to do something or if you want to encourage smone u can say. but generally people dont use it to comfort someone who is sad.