Can I create a contact in a particular contact group on Android 2.2? Is there a way to add a new contact into a particular contact group which I have in my Google contacts? Similarly, is there any way to move contacts from one group to another?

It seems as of now, I can only create contacts on Android but cannot put them in any specific group. To do that, I've to use the web interface of Google contacts.
Open the contact, click menu-->edit and scroll down.
You should see a drop down box titled groups just below the email addresses.  Click that and you can then select the groups the contact should be a member of.