"tonight" can mean evening or night, e.g. "see you tonight" would imply "this evening", whereas "I'll sleep well tonight" would imply during the night. Both are correct in English, but can be confusing! You could also say "see you this evening" if you want to; it means the same as "see you tonight".

The word in the exercise is shown on its own - without context - therefore knowing if they mean "evening" or "night" is impossible. If the exercise said, "see you tonight", or "I sleep at night", then there would be enough context, but it doesn't. Without context, both answers should be accepted.

You can say 'wieczorem' without 'dzisiaj'. For example 'Do zobaczenia wieczorem' - 'See you in the evening'. By the way, 'wieczór' means evening, so I don't think this is the best translation for 'tonight'. A better one would be 'Dziś w nocy.' (noc means night).

No, but you can say „W nocy będę spał”('I will sleep at night') or „Dziś w nocy idę na imprezę”('Tonight I'm going to party') or „Księżyc był wyjątkowo jasny tej nocy”('The moon was especially bright that night').