The first few lessons use a deliberately slow and careful/distinct pronunciation to get you used to Welsh.
As far as I know, "dwee" is common in actual spoken Welsh. (One Welsh person told me that dw i eisiau, four or five syllables depending on how you count, turns to dwisho (two syllables) where she is from.)
I suppose so, but I've also heard people say "we might say isho/isio but when we write, we should write eisiau".
They're not separate words, as far as I know; one is just a spelling that is closer to how that word is pronounced in some areas.
So I've understood it that some people write "Dw i eisiau" but say "Dwisho" -- i.e. they don't represent the -o ending in their writing even though they have it in their speech.