Either still an F-1 student or "out of status."

Can I travel to Santo Domingo with expired passport?

No - need a valid passport to travel overseas.

In form I-485, for non-immigrant visa number- Im an F1 grad student but my visa is expired. Do I still enter it in?

You cannot leave any box blank. Enter that visa number.

Can I have any no-payroll/no-employment period after the H1B transfer approval date but before the actual working start date?

Such a two week gap could only bring trouble ahead.

Confused- I-20 sent uscis with the original physical signature?

You will not face any problem. Wet signature is acceptable.

Can I use my I20 for Change of Status to apply for a visa outside the US when it's still pending?

You will need a new I-20. Make sure it will be correctly issued.

Is it legal to continue to do business which I started when I had a valid EAD, but expired now?

You need to stop working when EAD expired and can only resume one in possession of the new valid one,

No relevancy, no correlation to your eligibility for an F-1 visa.

I am a citizen of Jamaica, now living in Savannah Georgia as a permanent resident. How do l go about getting dual citizenship?

Any problem there might be lies with the Jamaicans. The US does recognize dual citizenship. Seek to clear it on the Jamaican side.

Really makes no difference but you have many other factors to consider and requirements to satisfy which I don't think you'll be able to if a...