If anything the opposite should be true. GPT-4 at least has near perfect English. If your sample displays non-native traits probably it wasn't generated by GPT!
I also wonder which proportion of English writing (in general) is written by non-native speakers, and whether we might be disproportionately represented in training data.
Had the training cutoff been prior to SEO and "content generation" farms, as well as a shift in balance of academic writing published, the embedding space would be different.
A low perplexity means the text isn't massively different from what it might have output itself (which might be an indicator that it was produced by a model), whereas a high perplexity suggests it's the kind of semi-random nonsense you'd expect from a student. ;)
Only if your prompts are themselves generic.
You can give it examples of a text whose style you wish it to emulate and it will do it.
You can also prompt it to alter what it wrote if you don't like something. For example, if you thought some part was too cliche and obvious you can point that out and have it alter that part.
If you have a back-and-forth conversation with it about what you like and don't like, what you want or don't want, with examples, the results can be much better than what you'd get with a generic prompt.
Incidentally, for creative writing I've found Claude to be much better than GPT4. I have not tried the version of Claude that's been enhanced with a 100k token context length yet, but that should allow you to give it many more examples and hopefully that will translate to even better output.
That said, none of these LLMs are perfect, nor do they yet pose a threat to really good (never mind great) human authors, but they're a pretty effective tool.