Splint on throwing thumb

Question
I have a player who needs to wear a splint on her throwing thumb. It is essentially a small rubber plastic brace. The doctor suggests that when playing she use a batting glove over top to ensure it does not move around. The brace is on her throwing hand (right hand in this case). Is this permissable in game play for this medical situation?
Response
Assuming the splint is properly padded and the player is NOT pitching, it would be allowed.
A splint may be worn, if padded. Rule 2.6.
There is no rule to prevent a player (other than the pitcher) from wearing a batting glove on her throwing hand.

Dropped Third strike

Question
Ball is pitched, Batter swings and misses for third strike, ball hits in dirt behind plate, catcher catches cleanly (no drop).
Is the batter out or can she attempt to get to first if not tagged or thrown out?

Response
It would depend on the situation. How many outs are there? Was First Base occupied at the time of the pitch?

If there are two outs or First Base is not occupied, the batter becomes a batter-runner who may advance to First Base.

If there are less than two outs and First Base is occupied, the batter is out.

See Rule 7 Sec 14 para D:
A batter shall be called out when:
A third strike (in flight) is not caught, provided a runner occupies first base at the time of the pitch and there are less than two outs.
NOTE: If there are two outs or if no runner occupies first base, the batter is not out unless the third strike is caught. The batter is entitled to try to reach first base before being tagged out or thrown out.

Circle on uniform

Question
I've been asked about our team logo on the front of our jersey. The logo contains a small yellow softball. I know in Little League we couldn't have a ball of any kind showing. I can't find anything under USSSA that addresses a ball or yellow circle on the uniform.
Response
A ball or circle that looks like a ball would be illegal on the glove.
We have nothing that addresses a ball or circle on the uniform.