Here's a video to watch what is happening:
I could really use some help, thank you in advance.

Dash issue
- When the car starts in the morning, everything works
- After a short drive (the amount of time varies) the chime will ding and the dash lights up plus the tach, speedo and water temp gauges go dead.
- After a random time (seconds, typically less than a minute), the dash lights will go off and the gauges will resume working.
- If I am parked (like in this video) the power locks are triggered.
- When the AC is ON and the warning lights go berserk, the cold air stops but the fan continues to run
- Sometimes the VSA/ABS lights turn on by themselves but I think that's a bad ABS sensor, but they aren't always on.
- The TPMS light is on because I have a bad sensor, I don't believe it is not related to everything described.
- If I stop the car and shut off the engine when the issue is happening, a restart "fixes" the issue until I drive for a short period of time (or let the car idle for a while)

Navigation Unit issue
The navigation display cycles through various screens
- Sometimes the "Enter PIN" screen appears.
- I have to be very quick to enter the PIN, but it typically (99%) locks up/resets during entry before it will display a map
- I have made it to the map screen once, but it was completely unusable.
- This issue with the navigation unit is preventing me from accessing the non-navigation menus (like setup)
- I am unable to display the "f-bus" diagnostic display because the navigation keeps resetting.

Random (now rare) No-Start
- I used to have a problem with the car simply not starting.
- The starter would not even turn the engine over
- I replaced the starter relay and it solved the problem, but it did happen one more time
- When the no-start condition was going on, holding the key in the start position for about ten seconds would eventually get the car to turn over.

I have not added after market parts to the vehicle.
A few yeas ago, I disconnected & removed the Hands Free Link circuit board.

What I have done...
- Replaced battery terminal connectors (not a fix)
- Replaced battery cable (kept getting blu-ish corrosion on battery)
- Checked all of the fuses (all were fine)
- Replaced the AC relay (it went bad before all of this started, but the relay was replaced after all of this started)
- Replaced the Starter relay (see notes above)

My thoughts... I don't think that I have a loose ground cable as the problem continues when the car is not moving. Also, see the last bullet point in the Dash section above.

I hope someone else has been through this and can provide some guidance.

 

Since problem affects multiple systems, I would either suspect power supply problem or CAN bus problem.

You wrote that you replaced the battery cable. I'm assuming both ground and positive cables. If battery and alternator is good (get it tested, ~14V while running doesn't necessary means alt is fine), then I would put scope on the can bus and check if it gets grounded / pulled to power / etc. while the problem starts. Maybe one modules on the network is acting up, or wire rubbed somewhere and is randomly touching the ground / other wires. For example door jamb wires, or water dripping on ECU from clogged moonroof lines (but here I'm just throwing ideas).

If you have access to bi-directional scan tool, you can scan all modules and look for lost communication codes, as they might get you in the right direction.

Overall, pretty hard to diagnose.