Are there any estimated, imperfect or fuzzy sorting algorithms?
The term you're looking for is approximate sorting. This is a topic that, as greybeard mentions in a comment, was more popular back in the 90s. But it's quite helpful for presorting (e.g. getting an array a little bit sorted to ensure that Quicksort doesn't get stuck in its worst case), and this is what's led to most of the research in the area.

You can, in fact, make it somewhat faster than full sorting. Giesen, Schuberth, and Stojaković showed that you can get $n$ elements sorted to a Spearman distance (*) of $\frac{n^2}{\nu(n)}$ in $O(n \min(\log n, \log \nu(n)))$. So the more you're willing to sacrifice accuracy (the smaller you make $\nu(n)$), the faster the algorithm can run.

Giesen, Schuberth, and Stojaković also present a very simple algorithm to do this approximate sorting, either deterministically or with a bit of randomness mixed in. The full details are in the linked paper, but basically they use Quicksort's partitioning a limited number of times and don't care about the order of the elements between the partitions, as long as they end up in the right "bin". If you pick your partitions well, this is in fact equivalent to running Quicksort and limiting the recursion depth.

(*) The Spearman footrule distance between two sequences is the sum of the distance between each element's intended position and its actual position. So if you swap two elements, that leads to a Spearman distance of 2: those two elements are each one position away from their intended place.