Is this a contraction of something? How does it differ from something like 寝ると or 寝てると?
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The ーと in this sentence relates two parts as a consequence, but also gives an admonishment: I.e. this character is effectively saying: "You shouldn't sleep in such a place or you will catch a cold, you know!".
(Contrast this with ーば (as in こんなとこで寝れば...）which would just states the fact/opinion plainly: "if you sleep in such a place, you will catch a cold.")
The "tto" contraction used here is emphatic, or possibly male speech. Or both. It strengthens the statement, as if the character is saying, "you really shouldn't sleep here."
It is a phonetic feature of Japanese that changes toward stopped consonants are emphatic. For instance consider：
「。。。」と → 「。。。」って
とても → とっても