Could it also be "Mujhe tel de do?" (sorry, do not have a Hindi keyboard at the moment)
Yes, but I think it's not proposed because this "verb +देना/लेना/जाना (and others)" hasn't been introduced yet.
तेल मुझे दो should be accepted or is it unnatural?
Its sounds unnatural to me.
तेल दो मुझे could work the same as original sentence though the emphasis would be on oil.
Actually in colloquial you can even just say तेल दो and make sense in the context, no?