Ben covered it well. Just a couple extra clarifications that often come up:
Likewise, if I am towing and successfully dock at my home island, does the derelict ship (after being moved to a docked position on the home island) then get an action on my turn?
Generally speaking, the ownership change from towing/capturing an opposing ship takes effect immediately. Because it is now a ship in your fleet, and hasn’t yet been given an action that turn, you can give it one. (Being derelict of course limits the specific actions available.)
Example 1) where you were towing and the derelict ship being towed had a shipwright aboard, could the derelict repair itself while being towed? Repair is not a move or shoot action.
When the repair happens, the ship is no longer derelict, so that breaks the tow immediately. If you’re looking to tow the derelict any further, you’ll have to make sure to move it with the towing ship before you start repairs.
Example 3) can your ship initiate a tow and then you immediately try to scuttle the ship you just started to tow?
You can make an attempt every turn while the ship is being towed, but if successful, the derelict still will not sink until your following turn.