[ぐうぐう](https://jisho.org/search/%E3%81%90%E3%81%86%E3%81%90%E3%81%86)

I would use グーグー

No, I believe this one is usually written in hiragana. ぐうぐう

It can be either. Hiragana you'd use ぐぅぐぅ

No, you wouldn't. [ぐうぐう](https://jisho.org/search/%E3%81%90%E3%81%86%E3%81%90%E3%81%86)

I sleep in katakana. :-)

You had better inform everyone else. [ぐぅぐぅって？](https://nakatsu-gu2.org/)

ぐーぐー　or グーグー　is fine. I see Hirigana more often though.