Well, to me "Would you mind calling me tonight?" is annoying long and polite and takes so long to say that by the end of the sentence I might forget the meaning. But that's just me and some people here have notions of politeness that I really don't understand.

You used it correctly, and the second statement is more polite. 'Would you mind' is a polite way of asking a person to do something. It is a way of changing a demand into a polite question. Using it is like saying "Would it bother you to..." For example, three ways of getting help tying your shoes: "Tie my shoes for me" this is a demand and can sound rude "Could you help me tie my shoes, please?" This is a question and is polite "Would you mind helping me tie my shoes, please?" Also a question, and polite

"Would you mind.....?" is used when you want to sound particularly polite, or if you think the answer may be negative. you can use it with older people or people you don't know well. the answer is like: No,not at all but " can...?" is to make less formal requests. you can use it with friends and people younger than you. the answer is : yes, certainly

Both of these questions are perfectly acceptable and can be used to ask someone if they would be willing to call you tonight. The expression "would you" is slightly more polite. Also when you ask "Can you call me tonight?", you may actually be asking someone if there are physically able to call you and don't have some sort of conflict that prohibits them from contacting you.

I think you can use them both but Would you mind is more polite ,you can use it when you want to ask something like would you mind pass me my bag , would you mind open the door, and also you can use Can you ,so both of them are correct , hope I helped you :)