If you say "yesterday", it suggests during the day-time. Otherwise you say "last night". I suppose "yesterday at night" is grammatically correct, but no one would say it.
You should always use 'tomorrow night' when refering to the night that follows this one.
'The next night' gets used sometimes when you are talking about the order of nights, rather than that specific meaning. (In the way that you might catch the next bus for example). However, people would usually say 'the night after that' in these situations, so 'the next night' is just an a