Even if Harry now believes in souls (and what happened to Hermione looks to me like magical death throws, if I was Harry it would be insufficient evidence to overcome my priors) then surely he would want to preserve her body so the soul has something to occupy? Maintaining the transfiguration is easy, keeping it cold is harder as you would have to repeatedly cast the cooling charm, but I doubt the internal structure of a strong metal changes much over time even at room temperature, and he would ask Quirrel for help, because adult help will probably be needed anyway for recovering her soul. Quirrel will be delighted to see Harry devote himself to the magical conquest of death.

As for time turners, being able to go back further than 6 hours and change history is an instant victory, unless both sides have them, in which case the story changes completely into a time war. I don't think that's where EY wants to take it, unless maybe in the last chapter, post-magical singularity, they travel back with invisibility cloaks and save everyone who has ever died.