I believe these three things are more or less interconnected. Currently the Hornet does not have a proper alignment process. So while the display might say you are aligned, you might not actually be. The best thing I’ve found to do when cold starting the Hornet is set the dial to align and then wait till the map appears on the AMPCD, ignoring whatever the text on the display says about alignment status before switching to NAV. Doing this seems to prevent the issues you mentioned for me.
You are correct with the shortened INS - the map itself doesn’t appear until the proper INS timing (you can set to NAV and start flying once the OK is up) is completed - when the short INS time was patched in, there was a report that the map still took the old longer time to show up and I don’t think that was fixed or changed.
The map also is not available at all zoom levels for all sections of each map, so try adjusting the range scale to get the map to display. There is more map sections for Caucasus than the Persian Gulf.
As for waypoints not being correct, that sounds like the INS switch wasn’t set to NAV or INS align wasn’t complete. You can also just do a baro altitude hold and fly level and straight - set INS to IFA for a couple minutes then back to NAV, just in case.