Yes, you’ll often get away with doing otherwise, but your return flight should be within 60 days. These days you can just go online and get a cheapo ‘fake’ onward ticket if doing it by the book is not your preference.
The best way to remember it is like ‘air’ but with a ‘g’ at the start. Transliterations should reflect the sound. ‘Ger’ just doesn’t. Anyway, up to you. I agree with you that ‘gay’ is completely wrong. ‘Po’ is simply nowhere near either btw. It’s ‘Paw’. I should probably stop there. 😆