Flight departed from the gate 5 min before scheduled departure time. Refund options

Most boarding passes will have two times printed, the boarding time, and the departure time. Boarding time is typically 45 minutes before the departure time.

Boarding is usually open for 15 minutes starting at the boarding time. If you do not get to the gate within 15 minutes of the boarding time, you are counted as a no-show.

In Heathrow, you have to be there a full hour before your flight’s departure time, or else the security gates will not even let you enter the secure zone of the airport. This happened to me once. Got stuck in London traffic, arrived during the fifteen minute boarding time, but the service counter for the airline had already closed, and the security gate scans the boarding pass to let you in. It would not accept my boarding pass. The airline counted it as a “no-show” so I had to buy another ticket.

I ran into this problem twice about 15 years ago with American Airlines – first time at the gate 10 minutes before scheduled departure, the second 15 minutes before, and each time already gone. Both times they rebooked me for later flights, but weren’t apologetic about it. Second time, it completely messed up my itinerary (I needed to connect with someone else arriving at the destination about the same time, for further travel by car to somewhere I didn’t have the address for, and everyone else involved had unlisted numbers. Disaster.) Oh well.

Several years later I was very interested when The New York Times wrote a story about the fact that this had become official policy of AA, and that it had finally become enough of a PR black eye (as social media grew in prominence) for them to stop it. But what was interesting was why they did it in the first place – because they found in focus groups/surveys that:

• travelers glanced at their watches as they pulled back from the gate, and this moment anchored their future impression/recollections of whether the airline was efficient and timely;
• travelers assigned blame for delays to the airline if they left the gate late;
• but not if the airline was delayed out on the tarmac, waiting for clearance to takeoff – then they blamed the airport, or just rotten luck.

So it made sense to game their impressions by always leaving early, and if the occasional traveler was inconvenienced, too bad.

So anyhow, we reached there 2 min prior but were advised that the flight already left and the gate was closed.

Say the flight “departure time” is 10:40:00

You simply can’t arrive very close to that time. You have to arrive before the “gate closes”.

That’s that.

The time the gate closes is indeed given on the boarding pass stub, and elsewhere in many places.

It’s usually ten, twenty or thirty minutes before the stated departure time. Perhaps 10:20:00 in the example.

Regarding your four questions:

1) You checked in, but, did not make it to the gate before the gate closed. This is totally commonplace. (Indeed, I’d guess that most folks who don’t make it to the gate in time, were already checked in.)

2) “Departure time” is neither literally the time the aircraft leaves the gate nor take-off time. So there’s nothing to “confirm”. “Departure time” is a nominal time. {Note too that FWIW, the airbridge moving away, doors being locked etc, are all different from literally “the wheels beginning to roll”.} The gate closes at a certain time (10:20 in the example) and that’s that. Everything after that simply depends on airport operations.

3) You can get that info on the various tracking sites, eg flightaware.com. But it’s totally irrelevant because (sorry for the bad luck) you did not make it to the gate by the “time the gate closes”.

(Interestingly, I’ve noticed that many/most boarding passes now only display the gate closing time, rather than displaying the “departure time”. I guess this is sensible but I find it confusing, personally!)

If you check your ticket, it will say something like ‘all passenger must be at the gate and ready to board x minutes before the scheduled time, or they forfeit their flight‘.
I have seen times required between x=10 and x=60 minutes, but never less than 10 minutes. In other words, if you miss this limit, it’s your own problem, and they don’t owe you anything.
For Alaska Air, it is here: https://www.alaskaair.com/content/travel-info/at-the-airport/airport-boarding-times

Boarding:

You must be checked in with a valid boarding pass at the gate
no later than 30 minutes before your flight. Being late may cause the
cancellation of your reserved seats and/or your entire reservation.

You typically can apply for a refund of the tax and fee part of the ticket price, as they haven’t paid those taxes/fees if you are not on the flight, but the remaining part of the ticket might be lost (as are all follow-up legs of the flight), depending on the airlines conditions.

If you ask at the counter right away and nicely, many airlines will get you on the next flight, for a small fee or even for free. But they do not owe you that.