Rule 109 - Signal and Timing Devices

A face-off occurs with ten seconds remaining in the period. As play begins, one of the On-Ice Officials notices that the clock has not started. Should the Official stop play immediately to remedy the situation.

No. Rule Reference 109(a and b).

As soon as the Official notices that the clock has not started, they must begin counting down the remaining seconds in their head. If they count down to 0:00 and the period should have ended but didn’t, they must stop play to end the period.
If the Official properly counts down to 0:00 in the above situation, but does not stop the play for whatever reason, any goal that has been scored after they count down to 0:00 shall be disallowed

Rule 617 - Goals and Assists 

The goal is tipped forward from behind as the result of player contact with the goal. The goal posts remain anchored to their position on the goal line. However, the cross bar pivots forward and down. Can a goal be scored while the goal is in this position?

No. Rule Reference 617(a).

The goal is considered to be displaced when the goal frame moves, causing the goal posts to no longer be on the same plane as the goal line (extended).

 Rule 402 - Minor Penalties

Both teams are serving one non-coincident minor penalty (on ice = 4 on 4) and the Referee is signaling an additional minor penalty to Team A. Play stops when Team B scores a goal. Does any penalty terminate?

The signaled penalty is washed out as a result of the goal, and the penalties on the clock remain. Rule References 402(c) and 409(b).

The teams were at equal strength when the goal was scored; therefore, no penalty that was on the clock would terminate. The delayed penalty is then washed out just as it would have been had both teams been playing at full strength.