"Zum Beispiel" is better translated as "for example".
That's also accepted, and would be the more common example without context. Here, I don't think it's necessarily better per se.
Can't one say 'I want for example to eat cheese'?
I'm almost sure the words order here is correct enough
This simply does not feel natural. There needs to be some indication that the "for instance" comes at the end.
What do you mean there needs to be an indication? Other variants are accepted with different word order. What would you find more natural?