Why ydy hi'n not mae hi'n here?
ydy at the start of a sentence is used to form a question (yw is not used for that). Mae is used to start non-emphatic statements.
This is explained in the 'Hints and Tips' for 'Present Tense 3', the section just before 'Weather'.
I thought it would be like this ('ydy' for questions), but the BBC big Welsh challenge actually have a scene where they say "Mae hi'n oer, tydy?". Does that work because it's a questioning statement or something?
Yes, that is a statement follwed by a tag question.
Diolch yn fawr, Ibisc!
Might this also be correct, "Ydy'n bwrw eira?"
Not really. Best to stick with Ydy hi'n...?.
Is there a reason that "mae'n" works instead of (as a contraction of?) "mae hi'n", but "ydy'n" doesn't as an alternative to "ydy hi'n"?
I tried on purpose typing "Yw hi'n bwrw eira?", and I got wrong, Why is yw here not allowed?