For the purpose of spells that target objects, not creatures, do victims of Flesh to Stone count as objects? I am especially interested in Shrink Item, followed by Stone to Flesh to get a less-than-Fine-sized character.
Creatures affected by Flesh to stone or similar effects (gorgons for example) are in a special case. For many purposes yes, they are objects, but for others, they still count as creatures.
In the given scenario, what would happen is you have a small statue that looks remarkably lifelike, but once they are returned to their fleshyness, they are no longer a valid target for shrink item and the spell is still in effect but isnt affecting them anymore. The lingering aura of the spell would only effect things like detect magic and dispel magic. Now, there is some confusion as to what happens if said person becomes stone again. Would they re-shrink because the spell is still in effect, or stay the same size?
I have asked a question about being able to alter creatures while they are stone.
Flesh to Stone only maybe makes objects
Read this answer.
To recap: the published text of Flesh to Stone definitely does not turn a creature into an object, but that creates huge problems so the creative director says instead it totally does make objects instead of creatures, actually, and because of Pathfinder's emphasis on said statements as a sort of pseudo-errata (and possibly because in this specific case he's right that not changing it has huge largely negative consequences) that's pretty much how it works.
Obviously, if there's no object, there's no shrink object.
This works otherwise
Spells don't stop working when you become an invalid target; nothing says that happens and lots of spells just would not function at all were that the case. Being a valid target matters when a spell comes into effect, so if you're an object when you get shrunk you would still be shrunk after a Stone to Flesh got rid of that petrification. Assuming you used to be Fine, you'd now be Fine-3. The spell could even be made permanent, and your body given a mechanicless clothlike composition via a similar sequence of events (adding in permanency and replacing Stone to Flesh with Break Enchantment).
That said, getting buff spells on normally invalid targets is pretty high-op stuff, and I wouldn't expect this to be comfortable for everyone at every table-- check about intended power levels before you start busting out stuff like this (and, of course, the abilities that reward you per size category for being small/smaller than your adversaries per category of difference)
Note also that going below fine or above colossal, while totally a thing that can happen, is not well defined in the rules. There's gonna be a lot of extrapolating formulae, or a lot of weirdness (e.g you either have a crazy-high stealth score because extrapolation or an effective size modifier of +0 because 'Fine -3' is not in the list).