Is there no sorting algorithm with all specific desired properties?
WikiSort and GrailSort are two fairly recent algorithms that do in place, stable, worst case $O(n\ lg(n))$ key comparisons.  Unfortunately I don't understand them well enough to know if they approach $O(n)$ swaps or are adaptive so I don't know whether they violate the fourth and fifth conditions you have.

From looking at the paper "Ratio based stable in-place merging", by Pok-Son Kim and Arne Kutzner linked to by the WikiSort GitHub page, Kim and Kutzner claim to have a 'merge' operation that is $O( m (\frac{n}{m} + 1))$ (WikiSort is a variant of Mergesort) but I'm not sure if that translates to WikiSort having $O(n)$ swaps.  GrailSort is claimed to be faster (in the WikiSort GitHub page) so I could imagine that it's likely they both have worst case $O(n)$ swaps and are adaptive.

If anyone does manage to understand WikiSort and/or GrailSort I would appreciate them also answering my open question about it