work authorization update

My H1 will be expired in 9/2011. I got my green card in 9/2010. I am going to stare my new job in 1//2011 in another company. Some questions:
1. Is is ok for me to change my job after receiving my green card for 4 months?

2. Should I fill out form I-9 to my current sponsoring firm to update my work authorization?My current does not know that I received my green card.

3. Should I show my green card to my new employer when I starts my new job in 1/3/2011?

4. Any other suggestions?

 

Thanks and Happy Holiday!