B. The motor for my mini lathe is a single phase 110 motor that has
wiring to allow rotation in both directions. I will not have a
controller. It will be wired directly to 110 AC through whatever
switch(es) I install.

1. If I flip the polarity on the switch to instantly reverse motor
direction, does the motor gradually slow down, then gradually speed up
in the other direction,

No, that requires a VFD controller.

or does the motor attempt to slam direction
changes?

It can't slam if it is a reversible capacitor start induction motor commonly
used on shop equipment like Grainger item 4k144.. The reversing switch changes
the polarity of the starting capacitor. When the motor is running, the starting
capacity is not energized, hence reversing it has no effect on the running
motor.

In other words, to avoid slamming direction change, I must
allow the rotation to stop before re-energizing the motor in the
opposite direction?

See above. In order to achieve reverse rotation the motor must be turned off
and allowed to stop then turned back on. When you reverse the capacitor makes no
difference.

2. If I flip the polarity on the switch to instantly reverse motor
direction, does this damage the motor?

How can it? See above.

In other words, to avoid motor
damage, I must allow the rotation to stop before re-energizing the
motor in the opposite direction?

No, to avoid motor damage don't hit it with an 8 pound sledge hammer. To
reverse the motor, see above.

Assuming the slamming effect is what happens and/or that motor damage
happens,

What slamming effect? There is no slamming effect with this motor unless of
course you are using the MOTOR as an 8 pound sledge hammer!

then I will use some sort of three position set-up where I
turn off rotation in one direction. Wait until the rotation stops,
then re-energize the motor for reverse rotation.

You can use a center-off three position switch if you like but it will be a
waste of money. A regular old DPDT switch will work just fine to revers the
capacitor. To turn the lathe on and off (regardless of rotation) just use the
switch that came with the lathe.

What I do is mount a toggle reversing switch in a little housing attached to the
motor wiring cover when the motor is accessible. See Grainger item 2x660, Motor
Switch Mounting Box

Dan