Is this form something that would typically be provided or completed at the check-in counter, or are travelers expected to print it out and bring it with them?
Depends on the airline. If they require it, airlines will generally have copies at the counter but they may be grumpy about handing them out (the darn thing is 5 pages long) . It's certainly easier to fill and print up front if you have access to a printer.
We are not US citizens or residents, and are just transiting through (train and bus and car into Mexico, same day)
That doesn't matter. The US does not have a concept of transit without entering the country, so there is no material difference between entering the country or just transiting. You will have to go through full immigrations and customs procedure at your first airport in the US.
Same for Canada. I was once asked by a Canadian Immigration officer: "how long are you planning to stay in Canada" to which I answered "hopefully only another 45 minutes". They didn't think that was funny though :-)