can teams sign new coaches even after roster-lock?
No, Riot does not make a difference between players coaches or analysts, the people who is inscribed during the window are the members of the team, all they can do is switch roles
They can't hire a new player/coach but can kick one. That's why kc is forced to play their assistant coach rather than getting 6th man
Because they already had a second coach to replace him with. What you can’t do is sign someone new, but you can tell someone you don’t want him to do anything anymore
"The General Manager (as defined below), Coaches (as defined below), and any Owner of the Team may not be on the Full Team Roster (as defined in Section 3.5 below)." - from the rules.