There is no hard and fast answers. Just remember the green card is for living and working in the US. With a travel pattern you have described an argument could be made by cbp that you have failed to establish the USA as your residence. Are you planning to apply for citizenship?
To my understanding the employment authorization is not renewed, you have to apply and be approved for adjustment of status. Then once approved you get permanent resident card and that overrides the eac card
It shouldn't be a problem. As long as the form is properly completed. I believe the form expires after 90 days from signing. I did this renewing my kids passport with no issues
Officially not longer than a year but try to keep it under 6 months. Otherwise you have to file for reentry permit. Also prolonged absence delays your ability to file for citizenship
I wouldn't do that because then uscis might reject the whole package for lack of payment. You have proof that it was received and that's all you need for now.
Fee's apply to 485, 765, 131 .previously all in one if sent with for 485. Effective April 1st Fee's paid separately and with separate checks. 129f fee already paid so no need to worry
Only include what is asked on the 129f form. Beneficiary passport not required. It only asks for last visit to USA and if applicable the details of entry if currently in USA. Text messages are weak evidence