Do airlines count the days for visa nationals?

Purely anecdotal & conjecture answer

1. Airlines rarely rummage through your passport to find an entry or exit stamp. That takes a lot of time in my passport, and I notice when it's happening at immigration or check in.
2. Airlines will sometimes collect or inspect your departure record card (if applicable)
3. They do look at the expiration date of your passport and the presence & date of Visa (if applicable)
4. Some airlines are indeed plugged into the immigration data bases of some destination countries. They will check if you are on a do-not-fly list. Another example: I once checked-in in Tokyo for a flight to Australia and due to me being stupid, my ETA (electronic travel authorization) had not enough days left in. The check-in agent flagged this immediately (and was nice enough to help me out). Since the ETA isn't physically in the passport, the only way for her to know this would be direct access to the Australian ETA system from her check in desk.

for countries/areas that have rules such as the 90/180 rules of the Schengen Area, do airlines bother to count how long you have already been in the area?

It’s not the airline responsibility to calculate those dates. That’s the passengers responsibility. It would be unfair to expect airlines to do that considering some passengers are frequent travelers with complex travel plans and histories. They are limited to confirming the validity of the visa, no fly lists, etc that’s it

For the United States APIS, this is the information required

• Full name (last name, first name, middle name if applicable)

• Gender

• Date of birth

• Nationality

• Country of residence

• Travel document type (normally passport)

• Travel document number (expiry date and country of issue for passport)

• [For travellers to the US] Address of the first night spent in the US (not required for US nationals, legal permanent residents, or alien residents of the US entering the US)

It is (normally) the airlines responsibility to confirm that you have the requirement documents to enter the country you are travelling to. It is NOT their responsibility to confirm that you meet the requirements to actually use those documents.

For example, if you are travelling to a country that requires a visa, and you show the airline a tourist visa, then they have met their level of responsibility around confirming that you have the documents required.

If you subsequently arrive at the border and state you are there to work, then you will likely be refused entry due to not holding the correct documents for your intended visit - but the airline will not in any way be held responsible because you did hold documents that would have, in a general sense, have allowed you to enter the country.

The same is true for time limits like the Schengen 90/180 day rule. The airline is required to confirm that you hold whatever documents are required to enter your destination country (which might be a visa, or simply just a passport from a country that doesn't require a visa). They are NOT required to confirm anything beyond that, such as whether you have sufficient days left in your 90 days for your stay - that is left to the immigration staff at the destination airport.

For some countries there is an additional step that the airline must carry out, which is that an electronic check must be done at check-in to confirm that the passenger is allowed enter the country. For example, all Australian visas are electronic, so the airline needs to electronically confirm with the Australian government that the passenger is allowed board the flights as a means of checking their visa. It's certainly possible that a country could enforce additional checks as a part of that process, but I'm not aware of any that currently do so beyond basic things like checking if the passenger is on a "no-fly" list.

There's not that much that an airline can do even if you did overstay your visa.

Generally, it's the airlines responsibility to make sure you are allowed to go from A to B, but not to make sure you had the right to be in A. At least, I think that would be the most common problem: people returning home after overstaying their visa.

But what could or should they do if some passenger arrives at the airport, after staying too long. Should they not bring them home? And if they do, who will fine the airline, and for what exactly?