Quick question, sudo-bash-steam

If I run sudo eopkg up; steam steam will run with sudo privileges or not?

No. The privileges end with the command, denoted by the ;

Example:
sudo id -u; id -u

0
1000

0 = root
1000 is my user (uid). Yours may be different.

Same thing when using && to string commands together. Permissions are not inherited.

Tank you