You are incorrect in your reasoning above. That's all I'm telling you.
Out of these equations: which one(s) are the variables not separable? Now I think II and III can be separated but I can not. Is this right? The way I came to this conclusion is by simply testing each one. I could not figure out a way to separate the variables for I. but for II and III I could. I might just be completely missing something for I and maybe it can be separate, maybe the other two can't and I just did them wrong! I'm not sure. As I said I couldn't figure out a way to separate I, but for II I had: d^2y/y*dy= 3x+dx after separating the variables and for III I had: dx/x=y^2/y-dy I'm not confident at all about how I separated them, it just doesn't seem right.
I think we've decided that II is second-order, so can not be separated. And III is first-order, so the question at least makes sense, but it can't be separated. So only I can be separated.
In your original post, you marked I only, but your argument was for II and III, the exact opposite.