"Your" understanding?

I have a bigger problem with "their numbers provided" than with the "your," to be honest.

In this case "understanding" is a noun. It's just like "Thanks for your patience".

Are there many signs in Canada or The States that say 'thanks' ibstead of 'thank you'? It sounds quite informal (to me ;).

The statement is correct with or without the "your."

It's true that most gerunds are seldom used with possessives (i.e. we say "thanks for coming" or "thanks for calling," not " thanks for your coming/calling") but "understanding" is an exception, and is very frequently used with possessives ("My understanding of the problem is that...")

I have a bigger problem with "their numbers provided" than with the "your," to be honest.

Yes. I assume the OP actually meant "not called in the order of the numbers provided." However, I've never seen such a sign in my life. What, then, is the point of the numbers?

What, then, is the point of the numbers?

If you just turn up at the doctor's, you have to take a number and wait your turn. If somebody comes in with an appointment, they get to jump the queue.

I didn't say it was wrong, just a tad informal.

i have seen such a sign before. The numbers are not called in sequence, when there aredifferentdoctors attending to their previous patients. So different doctors might finish sooner or later, but patients cannot go to any available doctors. Hope that made sense.

"thanks for understanding (v)" would be shaded more toward "thanks for reading and comprehending"
"thanks for your understanding (n)" leans more toward "thanks for your co-operation"

Neither is particularly elegant

Here in Chicago, the waiting room of a hospital near my home has a desk at the entry where you sign in and take a number. For privacy reasons you are called by number, not by name. There are several desks within the waiting room where admitting clerks interview patients individually and then send them to the appropriate doctor or medical department. As it happens, numbers are called in sequence. If those various clerks represented different physicians or separate medical practices, though, the next patient of Dr. Smith could have number 12, which might be called before my own number 8, because I am waiting to see Dr. Jones.

"thanks for understanding (v)" "thanks for your understanding (n)"

Really a gerund (verbal noun) in both cases. I'm not sure I agree that there's a meaningful distinction between the phrases, but I do agree that both are inelegant.