You're really talking about two different things. Configuring SSH to do key based authentication only is part of the sshd config, and sudo (passwordless or no) is handled by pam. What you should really be doing is this:
Configure SSHD to Only Allow SSH Key Authentication
Edit your /etc/ssh/sshd_config to modify and/or add as necessary the following
PubkeyAuthentication yes
PasswordAuthentication no
Then restart the sshd service.
Configuring sudo
You'll want to pick your sudo group and modify the sudoers file, using the visudo command and add something like this.
%wheel ALL=(ALL) ALL
That will allow everyone in the wheel group to execute any command as any user. Note that they will need to enter a password to do this. By default most distributions have them enter their own password, but sometimes sudo is configured to have them enter the target password.
Alternatively, you can configure sudo to not ask for a password at all. I don't recommend this except in the case of automated actions carried out by service accounts. In that situation you can specify only the service account can execute only a specific command. However, you're more than capable of removing the sudo password requirement by changing the above line to something like:
%wheel ALL=(ALL) NOPASSWD: ALL
