"This is a tasty cookie."
Translation:זאת עוגייה טעימה!
historically, זו is probably from Aramaic, while זאת is biblical. Neither is more formal of high-class than the other. Note that זו is pronounced "zo" - there's a holam there.
There is another word that's spelled the same, but pronounced "zoo" with a shuruq. That has a different meaning, similar to אשר. This word is not particularly high-class, but is pretty rare in modern usage.
Of course, not everybody pronounces the word correctly according to meaning, so you might hear "zoo" when meaning "zo". That is why I prefer to use זאת.
The word is זאת and it is pronounced "zot". That is the correct word and pronunciation.
Using זו and pronouncing it "zo" is a mistake, but a mistake that is so common that at some point people are going to accept it as a variation. The word זו should actually be pronounced "zoo", and it is an old-time alternative to "that", a synonym to אשר. But hardly anyone uses it like that anymore.
No. Spelling without nikkud doesn't have strict rules, and both עוגיה and עוגייה are common and acceptable.
However, if you follow the Hebrew Academy (which most people don't, at least not consciously), עוגייה is the right way: one 'י for the 'i' vowel and one 'י for the 'y' consonant.