Is it possible to see how many copies of a book are in the system?

Yes, its possible. If you put a book on your reminder list the number of copies in the system is listed.

I am sure there is a more sophisticated way and someone will come along any moment and tells us.

No, the only way is by reminder list.  There is no way to see where your book is in the FIFO line, though, but if you just posted it you can safely assume that your copy is last. :)

That's the only way - Reminder List. 

Use your ISBN to search then when it comes up click reminder.  I do a bit of weeding on my shelf that way.  I justtook one that had over 100 copies off.  Our church is having a rummage sale soon.  So that book is being donated.

I agree with Geri.

Geri -   you are a TG- congratulations :)