Don't have, don't need to and needn't mean essentially the same thing - you aren't obliged but you can if you want to. We usually use don't need to for general truths: You don't need to pay to go in that museum. But we usually use needn't to show that it's our opinion: You needn't buy me anything for my birthday. You could use either in each sentence but they sound more natural this way. We needn't bring our mobiles to school is a bit of a strange example. I think we don't have to bring sounds better.

Note that at least in US English I don't recall ever hearing anyone say "needn't" or "mustn't" except when intentionally trying to mimic a somewhat atypical style. These expressions are fine and correct, and we will understand them in the US, but I believe they are more commonly used in the UK. Of course, you may be learning English with a UK slant, and that's OK, obviously. I just thought it was worth knowing. I believe in US English, you are more likely to hear "need not", and more likely than that we would say "don't need to".

It's a difference between obligation ("have to") and necessity ("need to"). Those are two different concepts, but there is much overlap because people often feel obliged to do what they need to do, and likewise feel it is necessary to do what they are obliged to do. I need to sleep but I am not obligated to do so. I feel obliged to pick up after my dog but I don't need to.

Hi . I want to speak with you and improve our english if you have time ....on skype .or whats app .