How do "need" and "not" mix and match?

1. You don't need to play
2. You need to not play
3. You need not play
4. You needn't play
5. You need not to play

What does each of these mean, and which ones are equivalent to the others?

Is the meaning of the negation equivalent to what I would get if I replaced need with "must"/"have to"?

You should expand your contractions and keep track of what the word "not" is modifying.

1. You don't need to play => You do not need to play (here "not" modifies need)
2. You need to not play (Here "not" modifies play)
3. You need not play (Here "not" modifies need)
4. You needn't play => You need not play (same as #3)

1, 3, and 4 all mean "You are not required to play", whereas #2 means "You are prohibited from playing".

"need not" has the same meaning as "do not need" and the latter is more commonly used. Nobody would say #2 unless they meant to emphases that you need to avoid playing, and in that case they'd likely stress the word "not":

You need to NOT play (because if you play your head injury will come back and you'll be brain-damaged).

Even though #2 is technically correct people are far more likely to say

You must not play

to express the same prohibition.

Edit: #5, which was added, says

You need not to play.

This sentence is not grammatical to me as it stands.

I am interpreting needing to do something as meaning that something bad will happen if you don't.

1. This means you don't have a need to play. Nothing bad will happen if you don't play.

2. This means you have a need to not play. Something bad will happen if you do play.

3. and 4. These are the same as 1. "Need not" is an expression that means "You don't need". Needn't is an abbreviated form.

You can only replace "need to" with must in sentence 2. "You must not play" is similar to "You need to not play", and is a more natural way of expressing it. There isn't really a nice way of expressing a lack of need, using must.

You can replace "need to" with "have to" without changing the meaning, in sentences 1 and 2.

There isn't a good way of substituting "must" or "have to" in sentences 3 and 4.