Yeah, so this one threw me for a little, because the wording is really vague. What's throwing me is their use of the word "equilibrium". Given time, there is an equilibrium reached at ANY temperature. The interconversion between the two forms of tin is well known, and while the metallic form is prevalent at higher temperatures, the equilibrium tends to favor gray tin when it gets cold. (Gray tin is brittle and nonmetallic, and it has been theorized that the reason Napoleon lost to the Russians was because the buttons holding his soldiers' uniforms together were made of tin and didn't survive the cold Russian weather; that seems a bit simple to me as an explanation, but it's a fun idea that chemistry beat the Grande Armee...) There is a point, then, where the the equilibrium favors neither form, that is - where the concentration of each is the same at equilibrium. I have a feeling that this is what the problem means by equilibrium - that is, the "temperature at which gray Sn is in equilibrium with white Sn" is not referring to chemical equilibrium (where the rate forward equals the rate backward) but where the two concentrations are the same when chemical equilibrium is reached. Otherwise it really makes no sense, because chemical equilibrium can be reached for any temperature - the temperature just impacts the relative concentrations of the two forms when equilibrium is reached.
I hope that makes sense. And if my suspicion is true, that's an awfully poor choice of wording to use. I don't know who wrote the problem but if it's intended for international audiences, it's very possible it was originally written in another language and then translated, possibly by someone with little knowledge in chemistry.
Anyway, never mind all that.
What the question I think is asking is for you to find the temperature at which gray Sn becomes the dominant form, that is, at chemical equilibrium, when [Sn,white] = [Sn,gray]. Needless to say, in this instance the equilibrium constant K is going to equal 1.
So: At what temperature does K = 1?
I think you've figured it out from here. That's an easy justification for why ΔG° according to the problem solution is also 0. Notice if you calculate ΔG° at 298.15 from your values, you do get an answer very close to 0 (-0.007 kJ/mol), but be careful here because every source I have lists the ΔH0 for gray Tin as around -2.1, not +2.1, AND when you calculate ΔG° this way you need to use 298.15 K, not the mystery temperature you're trying to solve).
I think that will give you the right answer (using your numbers I got about 286 K, which is close to the experimental value; interesting when I use values from the CRC, I got a value that's way off. Another indication this question is bad.), but I've got more issues with this problem beyond those I've already described. I hate problems that rely on you making assumptions that you maybe wouldn't otherwise expect to make, but this is what this problem is requiring. You are also implicitly assuming that ΔH and ΔS are independent of temperature. So the ΔH and ΔS values you calculate from the heats of formation are appropriate at all temperatures. It's generally not a bad approximation, but good questions make it clear when you are supposed to make certain approximations. Add that on top of the poor wording and you get a problem that's really confusing IMO. I'm still confusing myself writing about it so I think I'll quit now.
In the end your logic was sound - it was the question that was screwy.