Do airlines have to check the visa status before selling a ticket?

Some countries require you to purchase a return ticket before issuing a visa. So no, the airline won't check your visa when you purchase the ticket. I am surprised The visa wasn't checked when she boarded the flight in Dallas

Short answer to an easy question: the airline is not responsible for your legal issues regarding obtaining a visa. They don't have to jump in as a consultant when you purchase a ticket. Obtaining a visa often implies, among other requirements, to already have a ticket, so, if the airline required a visa, it would be impossible to obtain a visa without a ticket and a ticket without a visa. That would make traveling impossible.

An airline is not required to check your visa and entry compliance before selling you an air ticket. You can buy a ticket for any destination even when you do not have a passport.

Check-in staff normally checks your visa before they give you your boarding pass and the first flight was to Dubai where your daughter did not require a prior visa so the airline staff would have issued a boarding pass and I don't blame them. When your daughter turned up at Dubai airport for check-in, that's when the airline staff checked her visa for India as that's what the destination was and they didn't let her fly because she didn't have a valid visa.

I do not think airline staff/airline was at fault. Had this been a flight where you through check-in your luggage to the final destination and get both the boarding passes at Dallas then it would have been Airline's fault.

In many cases it is impossible for the airline to require a visa or to check for a visa prior to selling you a ticket.

Tickets can be purchased up to 330 days in advance of travel, but many countries will not issue a visa more than 60 days ahead of arrival.

Many countries require proof of travel in the form of an e-ticket or confirmed itinerary before they will issue a visa.

As others have already discussed, the airline did exactly what was required of it. The burden of responsibility to make sure their personal documents are in order lies solely with the traveler. It is not the responsibility of the airline or tour company or hotel or travel agent.

You are shouting (unnecessarily) and blaming Emirates for what is your fault. Emirates provide lots of very helpful visa advice, such as:

Please double-check foreign entry requirements for travellers from your country of nationality

from their website, and their Terms and Conditions (in this respect effectively identical to all other passenger airlines) include:

Passports

Please insure you have a passport valid for the duration of your trip. (Some destinations require your passport be valid for a certain length of time; in general, 6 months after completion of travel). You may also be required to obtain a visa for some destinations and it is your responsibility to do so.

When making a booking you must ensure that your name (and the name of any person(s) for whom you are making a booking) matches exactly the name as it appears in your (or the person(s) for whom you are making the booking’s) passport.

If you require any further information on passport or visa requirements, please contact one of our local offices or the embassy of the country to which you will be traveling. A failure to travel with the correct documentation could result in the passenger being refused travel, entry to the country of their destination or stop-over, deportation or incarceration and in such circumstance(s) you will be solely responsible for any cost, loss or damage which you or we (or our agents or servants) incur.

My emphasis.

Emirates did you a favour by picking this up in Dubai rather than have the passenger refused entry in India.