Caught - Not out

The hand holding the bat, or any glove on that hand, is considered "part of the bat" for determining whether the bat strikes the ball:

5.6.2 contact between the ball and any of 5.6.2.1 to 5.6.2.4

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5.6.2.2 the batsman’s hand holding the bat

5.6.2.3 any part of a glove worn on the batsman’s hand holding the bat

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shall be regarded as the ball striking or touching the bat or being struck by the bat.

By omission, therefore, it does not count a hand or glove that is not holding the bat.

To be out Caught requires the bat to touch or strike the ball, among other conditions:

33.1 Out Caught

The striker is out Caught if a ball delivered by the bowler, not being a No ball, touches his/her bat without having previously been in contact with any fielder, and is subsequently held by a fielder as a fair catch, as described in 33.2 and 33.3, before it touches the ground.

Therefore, the batsman is not out Caught.

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Obstructing the field - maybe Out!

A batsman who willfully plays the ball with a hand holding the bat, except for specific circumstances listed in the laws, has met the conditions of Law 37:

37.1 Out Obstructing the field

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37.1.2 The striker is out Obstructing the field if, except in the circumstances of 37.2, in the act of receiving a ball delivered by the bowler, he/she wilfully strikes the ball with a hand not holding the bat. This will apply whether it is the first strike or a second or subsequent strike. The act of receiving the ball shall extend both to playing at the ball and to striking the ball more than once in defence of his/her wicket.

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37.4 Returning the ball to a fielder

Either batsman is out Obstructing the field if, at any time while the ball is in play and, without the consent of a fielder, he/she uses the bat or any part of his/her person to return the ball to any fielder.

The exceptions are listed in Law 37.2 Not out Obstructing the field, and amount to

• accidental obstruction

• avoiding injury

• the striker protecting their wicket, provided it doesn't interfere with a fielder trying to catch the ball

Therefore, the batsman may be either out or not out, depending on the circumstances.