Take a look at
this post for #2. Cars built after Oct 06 (read post for details) have TPMS built into the wheels; those before use another method.
TPMS will show an error in the instrument cluster when the warning is active. Otherwise there is no indicator. I don't believe pushing it when the warning isn't active is going to show you a thing; then again I don't go pushing it much except to clear an error or after adding air. Car isn't with me, or I'd go look.
Every now and then mine loses synch with the monitoring system and I have to press the TPMS to clear the error light (after I check the tire pressures of course).
Some of these questions are probably answered in the manual. Here are few places to find it if yours didn't come with one--I don't have a lot of the details memorized so I find myself referring to it from time to time.
Link