Does have a user in group 'root' all the rights as the user 'root'?
 

All know there is a user root in group root. Now I will create a user in this group. I think this user should have all the rights as the root. Right?

Not all of them. Users are not the same as groups.

You'll only be able to additionally read/write/execute files in the group root.

Root is a special user with "Super User" abilities as standard, if you want users to have similar abilities there is already the su and sudo facilities available for this task and these are generically speaking... safer then root. Sudo is a very very useful facility in this respect. Adding a user to the root group only matters in response to file permissions only what generally speaking root does not abide by anyway.

The background of my questions: I will install/deploy some programs which will access files from other users. So I want to create a user in the group of root.

Best to create a dedicated group and add the users & apps to that group. Don't add people or progs to root (even as group root) unless you can't avoid it.
Note that people can belong to more than one group at a time.
You can use the 'id' cmd to see your uid, gid and group memberships.