Basically, it is your own responsibility to check for the validity. The travel agent simply books you the for travel. However, she should be asking you if all your travel documents are up to date and valid for the travel.
he is a travel agent on the internet
Risky. Ask your travel agent or UAE embassy.
A passport, immunizations - check with your travel agent they will have a list of what you need to do to prepare.
You would have to be registered to be a legal agent or you would have to work in a legitimate passport agency or regional passport agency.
I would think not on the passport. All airports (to my knowledge) require an ID. Since you're going from one US state to another, I don't see the need for a passport. I'd call your airline or travel agent just to be sure.
For current information on what a country requires, you can talk to any airline telephone agent, or travel agent, or call, write, or email the consulate or embassy of the country you are going to be visiting.
Nothing because a travel agent and a travel agent are the same thing.
If it is a full British passport (not an overseas or temporary passport), then all that's needed is a passport for travel to other European Union countries. The passport can even be expired. However, airlines may have different rules; check with your airline or ferry company first.
you can do so as long as you have all your legal documents
Likewise, you do not have to be certified as a travel agent to become a home-based travel agent. Now there are several travel agent certification programs out there, most notably those operated by The Travel Institute, but that is something else entirely. Seeking travel agent certification and becoming a certified travel agent is a matter of professional pride, not a must-have to become a travel agent.
Certain countries require that there be a minimum time left on your passport before you can enter the country (eg 3 years to expiry) As these are generally countries ther require a visa to enter, the information should be given to you by the travel agent or particular embassy. But I don't think you can travel if your passport will expire before your return date unless you're an EU Citizen within an EU state because then there are ways to get a new one from home.
A travel agent is someone who can book your travel details - flight, hotel, transportation and even tours, based on your requirements. Their responsibility is to organize your entire vacation, or as much of it as you like. A tour operator is just someone who operates a business that provides tours of a certain area or series of attractions. They will organize and schedule tours for people, and will provide the guide. Their responsibility does not normally go beyond that.