How long would it take to have interview after employment authorization card received?

approximately 7- 8 months from the date of receipt notice. No problem so far. If concerned, check with uscis infopass

Can a nursing home facility sponsor or petition someone with an expired student visa?

Consult with attorney about details. In general, if you are out of status you cannot change it in US.

Can my wife take her annual 5-6 week vacation to Indonesia after taking the biometrics.?

Yes it shouldn't be a problem. Make sure to mention the exact dates of her trip to the officer at the interview.

I lost the i102 receipt, I just have the email confirmation and I would like to apply for the green card (i485)?

https://i94.cbp.dhs.gov/I94/#/home#section and view travel history. You need passport number. Include the history list in your green card package

Do I need to Report DHS about my departure?

Are you leaving US permanently? You have to consult with an attorney to discuss details of you situation.

Is it normal for my online adjustment of Status profile to remain unchanged after attending interview in a USCIS field Office?

Yes, it happens. Updates may take longer than you expect.

When was your first time you entered US? If after April, 2013 you can get your travel history and obtain current i-94 on...

yes, If divorced you become F1 category.

He cannot legally force you to sponsor him for the green card. However, an affidavit of support can be easily enforced in the court of law. EVEN...

Found my old green card but filed i90 and gave biometrics, can i stil ise old green card?

Your old green card is valid. You can use it for travelling, but you can't use both cards.