I think "cette" can mean both "this" or "that." But if it can't, then I'd love to know why!
It usually can but I think it may be skewed to that due to "they were worn out". "After this long day they were worn out" sounds odd to my ear.
"After this long day, they were worn out, but I was fine." This is perfectly okay!
If I am only hearing this, how can I distinguish between "il etait" and "ils etaient"
It's easy for this verb, because the liaison creates a "z" sound between the words (i.e., the "s" in "ils" is not silent). But most of the time, you can't tell the difference, except by context.
Why not accept "exhausted" DL if it's given as second meaning?
Turns out exhausted is accepted.
My answer "after this long day, they were exhausted " was rejected.
In the quicker audio there is definitely a liaison between ilS and étaient. If you listen in slow mode it is absolutely IL était. If you write the single verb, it's wrong. I wish the slower mode would be accurate to the translation Duo wants.