Some questions about b/g standard

After reading the sticky thread about basic conception of wireless network,I notice that b/g capacity are most widely used standard.But I got some questions which I'm still not very clear.
1.My wireless card is a b/g compatible card,which means it can pick up both the 11Mbps and 54Mbps wireless signal.But it still depends on your router device,right?
2.Does my wireless card automatically detect wheather the signal is b or g?If so,will it automatically shows on the systray icon that it's 11Mbps or 54Mps?Or do I have to set up different network connetion for different capacity?
3.In my university,the speed is fast but my wireless connection still shows that 11Mbps so that it means the wireless broadcasting is still a b signal,right?

Sorry for the stupid question,but still hope someone kind enough to help me out,thanks.

 

1.) Yes
2.) Yes
3.) Yes

Not stupid questions.