How do Chinese characters foreigners or beginner Chinese learners write appear to natives?
Weird yet right.
Is there a difference? Does it depend on how long they have been learning, or are they all different from a native Chinese's handwriting?
Knowing the basics is what it takes to achieve writing Chinese. Lines and curves are the most important thing to keep in mind. There are different writings in Chinese, just like English. Cursive is one of them. Time is a huge impact. Have you heard of this saying: Practice makes perfect.
Would the handwriting even strike a native as a foreigner's penning at all? And how about a beginner's handwriting?
Shrug. Foreigner penning and beginner penning is almost like the same thing but with different practice. Even a Native would need to begin with the same method of writing. I know because I'm Chinese and learn this from painful writing experiences like writing a word 500 times before I got it right.