Transporting a model category structure along a left adjoint

Instead of Theorem A.3 in the Schwede-Shipley paper, it should be Lemma 2.3. They prove that it suffices if all objects in $M$ are fibrant, and if every object in $N$ has a path object. I think the other references are not necessary. Probably the numbering convention had to do with an older version, where section 2 was an appendix. By the way, this result you mention by Berger-Moerdijk has been strongly generalized by Johnson and Yau in their paper on model structures for PROPs, and also in their 2nd book. A summary of Yau's work in this direction can be found in Theorem 5.7 of this paper of mine with Yau. As far as I know, that's about as general as you can make it.

My go-to reference for inducing model structures along an adjunction is Hess, Kedziorek, Riehl, and Shipley's A necessary and sufficient condition for induced model structures, which works in great generality.

It suffices to dualize the proof of Theorem 2.2.1 in Necessary and sufficient conditions for induced model structures. It uses indeed an argument coming from Quillen's book "Homotopical Algebra", II page 4.9 (the diagram in the bottom part of the page). And it is also necessary to use the fact that the class of weak equivalences satisfies the 2-to-6 property.