That would be the LEM IMU update using a theodolite and a survey point to convert to lunar units.
No, that's not required.
The LEM doesn't know where it is or what it's alignment is on the moon...
It doesn't need to know where it is on the Moon to any great degree of accuracy. The IMU alignment is to correct for IMU drift, not to establish the launch site location and spacecraft orientation.
The big problem is that there's no survey marker because somebody would need to have been to the moon previously to put it there. Saturn 5 had a glass window with a poro prism so you could see the IMU and the center of inertial reference and tell the guidance computer where the IMU is located in geodetic earth coordinates (to 5 decimal places.) It takes a guy on the pad with a theodolite and a plumb bob. They do this repeatedly up to the moment of launch.
And that's how you confirm IMU alignment when you're launching from Earth during the daytime. What makes you think that's the only way it can be done?
When lifting off from the moon, both the LEM and the CM would need to be running on moon coordinates so that the CM (moving much faster than a speeding bullet) could tell the LEM it's coordinates and velocity (state vector).
No, that's not how it was done. That's the hard way to do it. After the LM lifts off, the only thing that matters is the vector from LM to CM and the velocity along that vector. It does not matter in the least what lunar surface coordinates the LM left from. The CM does not need to "tell" the LM its "coordinates and [relative] velocity state"; the LM is perfectly capable of acquiring that information itself via rendezvous radar.
Clear you are unaware of how orbital rendezvous really works and the basic principles of terminal guidance. How you can claim to be a rocket engineer is beyond me. You're completely clueless.
And the LEM would need moon coordinates to know how to maneuver for rendezvous using it's known position and alignment on the moon, and calculating the intercept point in moon coordinates. There's no way around it. They would not have been able to do an IMU alignment on the moon.
No, that's not how it was done. You suggest that orbital rendezvous is a one-shot deal that depends heavily on terrestrial coordinates. In fact orbital rendezvous is a stepwise procedure using terminal guidance procedures that are completely removed from terrestrial launch conditions.
Some rockets still use the theodolite system prior to launch.
And some do not. So why do you suggest that's the only way to align an IMU?
You come here pretending to be an engineer familiar with launch operations, but clearly you aren't. You fail to understand basic physics, and have no clue how launch operations, guidance, and navigation are actually done. You can't even demonstrate a competent knowledge of how Apollo claimed to do it. You base your argument solely on your patently false claim to be an engineer, arguing from some semblance of personal knowledge and expertise. You seem to believe we should accept your claim to expertise
a priori as a given fact. That's not how it works. Instead we look at the ignorant arguments you make and from them conclude that you can't possibly be a competent engineer.