"Should I get 12v current at the starter output, at the thermal time switch and at the cold start valve while cranking the starter on a cold start?"

I believe yes.

"On a warm start I suspect I should get 12v at the starter output and the thermal time switch, and no current at the cold start valve. Is this a correct assumption?"

I believe yes again.

"Should there be any current (at starter output, thermal time switch or cold start valve) when the starter is not engaged (while ignition is on, or car is running)?"

I believe not.

Note: You may want to remove the cold start valve from the intake, connect the wires, and spray it into a can while cranking the (cold) engine. I once found a bad nozzle this way. (NEW valve doesn't always mean GOOD valve.)