The call?
 

Team A linesman has his mouthpiece swinging during their 2nd down running play. Flag is thrown, play continues & Team B recovers the fumble.

What do you have (Fed)?

You should have nothing but a warning for the offending player, since you need to see if the mouthpiece was hanging loose or in his mouth at the snap. If it was in and came out during the play there is no foul.

If you see the thing hanging loose you should warn the player to put it in his mouth before the snap to prevent a foul. Officials should try very hard to handle this situation using preventive officiating, this foul call should be discouraged. I'm not saying ignore it. If you throw a flag before the snap, you've got delay of game. After the snap you've got failure to wear required equipment during a down.

Handle it before the snap and you avoid a foul, or limit the penalty to 5 yards.
After the snap it's a 15 yarder.

Appreciate the course in preventative officiating, BUT can I get the original question answered?

Apparently the mouthpiece hadn't been seen swinging prior to the snap (freshman game w/3 man crew 1 no show) thus the foul after the snap but before the fumble recovery by Team B.

REPLY: Just to review...failure to have the proper equipment, e.g. not having a mouthpiece, or wearing illegal equipment is a 15-yard penalty. Having the proper equipment but not wearing it properly during the down is a 5-yard penalty from the succeeding spot.

1.5.6 Failure to wear required equipment during a down is 15 yd . Failure to properly wear equpment when ball is about to come alive is 5 yds. If you want the answer to your question, no they do not have to decline the penalty to keep the ball as this is a USC foul, succeding spot enforcement.

How can you call failure to wear equipment if you don't know when it came out, during the down. If it is out before the down starts you should then whistle if you are going to call this not after play starts.

Like others have said, this foul needs to be prevented from even happening, but failing that it must be called before the snap. If the snap has already happened, then you need to waive the flag off because it now appears you don't know if it came out before (which is a foul) or after the snap (which is not), and if you are not 100% sure about the foul you shouldn't be throwing your flag.
But if you insist on enforcing this, it's B's ball after a decline, unless of course they don't want the ball.

First off, why does Team A have a linesman?
 

I'm with the officials, not Team A:)

I believe the case book says that IF you see a player intentionally not using required equipment during the down-such as spitting out his mouthpiece or unbuckling his chinstrap it is a foul. If you do NOT see the act, then you are to assume the dis-use (?) resulted from game action and is not a foul.

If you catch it during the dead ball period, you should try to get the player to correct it and if not, flag and kill it for Failing to Wear required Equipment, which is considered delay of the game, I believe.

Right, to answer your question: Team B will decline the penalty and keep the ball. No problems here.

Just a simple question here. Does B have to decline this penalty to keep the ball? If this was a standard previous spot enforcement then yes they do. For most, if not all other succeeding spot enforcements they don't. Why would this one be any different?

You didn't read carefully enough. Seeing it dangle during the play is NOTHING. You MUST assume it came out after the snap since you didn't see it before the snap.