How long does it take to hit a unit's record books when switching from Reserves to ARNG?

You are "on the books" with an effective date of 28 JAN, as a 92A carried excess in HHT.
Assignment order was published 22 FEB 2021, you probably need to provide that and your guard contract to the reserves to get discharged from the reserves if you have not already done so.
You can pull that from IPPS-A self service-orders, or probably iPERMS by this point.

It's great that you are talking to your platoon SGT, but it looks like this is a case where you should have been talking to the AGR directly and keeping your platoon SGT in the cc line, because it looks like the issue resolved a while ago.

I recommend that you contact your AGRs, and provide them with an SF 1199A if you have not already done so, and remind them that if you are not showing up on the payroll, they can manually add you until you do show up. I recommend stopping by the unit in person, probably worth the 20 min drive.

You also need to get your DTS released from the reserves (help desk ticket on Trax) and if you have a GTC, just tell the AGR, that can be pulled over with just your SSN.

Depends on the unit, and how fast they push your paperwork to G1.

Service transfers are typically the responsibility of your state's G1. Your gaining unit has to accept you during the process but overall it's a state run process and if you aren't on the books yet, then I'd reach out to the contact who took point on transferring you as they're going to know who to bug about finalizing the paperwork to get you drilling.

I'm not sure how California handles that process, but in smaller states like mine it's typically a combination of a Recruiting and Retention Battalion person and the state G1. Hopefully they can get it ironed out for you soon.