If you set up public key authentication, you can do it with a passphrase-less key if desired, but do note that this reduces security as anyone who can get their hands on the key file will be able to log in to your account (subject to any restrictions on key usage listed in ~/.ssh/authorized_keys). Depending on your actual usage and situation this may be a serious problem or it may be no problem at all.
When you set up a session for PuTTY, under Connection -> SSH you can specify a remote command to execute. There should be no reason why this could not be
It might also be possible to configure
sudo to not require a password (for your account, or globally), but this will of course mean that anyone who can get their hands on a connected session will have complete access to the entire system just by doing a single sudo.