618 - Handling the Puck

The puck is batted forward with the hand, hits the shaft of the stick of the player batting the puck, then goes directly into the opposing goal. Does the goal count?

No. Rule Reference 618(a).

The deflection off the stick does not alter the fact that the puck was propelled into the goal by the hand.

502 - Referee

Should the Officials stop play immediately or allow play to continue whenever a player on the ice is body checked and crashes through a door or through the glass?

The Officials must stop play immediately. Rule Reference 502(a).

Anytime there is a potentially dangerous situation where a pane of glass has been broken or a door (other than players’ or penalty bench) has been opened during play, the officials must consider the safety aspect and stop play immediately.
The exception is when a player is knocked into an open door on the players’ or penalty bench and is able to resume play immediately and the door is properly closed.

601 - Abuse of Officials

A player is assessed a minor penalty for tripping and does not proceed immediately to the penalty bench while disputing the call. Should any additional penalty be called?

Yes, in addition to the original tripping penalty, the player must be assessed a minor penalty for Unsportsmanlike Conduct. Rule Reference 601(a.1).

The misconduct penalty (601(c.4)) for failure to proceed directly and immediately to the penalty bench applies in all other situations. In this instance, the fact the player is challenging or disputing the ruling of the Referee calls for the unsportsmanlike minor penalty to be assessed.