身を切るよう な寒い 中 で わざわざ 遊ばなくて もいい じゃない。
asobanakute comes from asobanai (i imagine), but why the jyanai? I know that the phrase mean "You don't have to play outside in the biting wind" but the jyanai negating that the person don't need to play on the cold only makes sense if the asobanakute don't have a negative meaning or is if adjectivating or serving to the moii in any sort of way.
No. It is not a double negation. The last bit じゃない, is kind of an interrogative similar to "right?" or "isn't it?" in English, with which the speaker looks for the listener's approval:
身を切るよう な寒い 中 で わざわざ 遊ばなくて もいい。You don't have to play outside in the biting wind.
身を切るよう な寒い 中 で わざわざ 遊ばなくて もいい[ん]じゃない。You don't have to play outside in the biting wind, have you/right?
Also note that connecting いい directly to じゃない is not really grammatical, even though you can find it in very informal speech. A ん or の is necessary to nominalise the whole clause ending in いい.
Finally, if it were a true double negation, you would find the i-adjective いい conjugated in the negative:
いい → よい → よくない
身を切るよう な寒い 中 で わざわざ 遊ばなくて もよくない。Even if you don't to play outside in the biting wind, it's not ok (note that the sentence makes no sense semantically).
You're right, 遊ばなくて comes from 遊ばない. Here it's being used in the grammar pattern なくてもいい, which in a literal sense means "Even if you don't X its ok"
The じゃない on the end is being used to create a tag question and essentially is the speaker seeking confirmation about their statement.
So your sentence in a more literal sense is something like this:
Even if you don't deliberately play in the piercing cold that's ok, you know that right?
