I am interested in finding out how swinging a baseball bat or similar object would affect the astronaut's rotation in free fall, on the ISS for instance. How much would the astronaut swing the bat, and how much would the bat swing the astronaut? Would the bat strike an object with the same amount of power as it would if the astronaut was standing on the Earth's surface? Would it strike with considerably less power since rotating the bat would also rotate the astronaut in the opposite direction?
How much would the astronaut swing the bat, and how much would the bat swing the astronaut?
In terms of angular momentum, the bat and the astronaut will be exactly equal and opposite; and in terms of angular velocity, this will be the ratio of their moments of inertia, which I'd guess would be something like 1:1, so these would be approximately equal and in opposite directions.
Would the bat strike an object with the same amount of power as it would if the astronaut was standing on the Earth's surface?
No. (I'm assuming you mean that the object is stationary in the initial frame before the astronaut swings.) There are two reasons for this. The first is that the bat is traveling at approximately half the speed (that is, it will be the same speed relative to the astronaut, but from the answer above, in the stationary frame so it will be half the speed). The second is that the astronaut has less force/torque behind bat at the time of the connection, which is important for the force the bat delivers. To illustrate this consider two examples. The first is a slap vs a punch, where a slap is approximately just the momentum of the hand, and the punch is delivered with the whole body behind it, including force from the feet against the floor. The second is pushing a car, where instead of running up and slamming one's body into the car, one carefully places their feet against the ground and pushes. Batting on earth is like this but faster, and losing connection to the ground will reduce the force. (The other way to look at this is the torque that's applied to the bat at the time of the hit, where in the space-case the torque is against the astronauts angular momentum, and on the earth case the torque is applied against the batter-earth system.)
Would it strike with considerably less power since rotating the bat would also rotate the astronaut in the opposite direction?
Yes, as I explain above.
Considering only physics of it and not body anatomy and the associated inefficiency of swinging a bat while rotating,
The momentum imparted in both the case will be exactly the same because of the fact that in both cases (that is while on ground and when in space) the amount of force applied on the bat by you is exactly the same. Dont get confused by the reaction force, the reaction acts on our body and not on the bat and the reaction is always there irrespective of where you are.
Put the batsman on an office chair on a nice smooth floor perhaps kneeling and swing and wallop away. I think that’s a fair simulation of horizontal motions for the problem. Curiously I think energy rather than momentum kills this! And angular stuff just complicates but gets the same... In free case, on the chair a small part of the energy released gets wasted on recoil of man and chair while with man on earth the recoil energy is negligible if he plants feet well and doesn’t fall back. Like in Stranger Things you need to plant your feet. But it is anyway just a small difference as like with a gun most energy goes to the bullet no matter what.