Can the 'ei' be omitted and still be good Welsh?
Yes definitely. Beth wyt ti'n wneud? and Beth wyt ti'n gwneud? are both absolutely fine in colloquial language.
Why the possessive pronoun here?
It refers back to the question 'what', literally 'what are you it doing', more correct English = "what is it you are doing"
My course book teaches 'Beth wyt ti'n wneud?' Not with the 'ei'.
Yes, that's more normal in colloquial language. Including the ei makes it more formal.
People go even further - in rapid speech the same sentence can sound like 'be ti'n neud'
If the ei can be omitted why does it mark it as incorrect?
It shouldn't. Report it. The mods are pretty good at keeping on top of things here.
"beth wyt ti'n wneud" is one of the 17 correct answers here