Intermittent Issues with Starting/Battery 2007

My car will sometimes be completely dead. Other times, I can start the car and it will either have been ok while off or will show evidence of having no voltage at sometime since I drove it (lost presets, on-screen info for miles and mph zeroed out). Consistent with this, there have been times that the car would not start, I drove my backup car, and later - sometimes days later, the car will start right up. My band-aid solution is a portable charger, all I have to do is connect the charger to the charging points and it starts right up. However, about half of the time it starts, it goes into accessory mode automatically without me even entering the car. My neighbor knocked on my door the other day as my wipers were on (I must have bumped the wiper arm when getting out after the car wouldn't start). This last time, I was finally able to get my voltmeter on the battery and measured 0.9V. (I would like to have words with whoever designed the way to open the hatch without car having power, but I digress)
I did replace the 12V battery and the problem continued. I took the battery to the auto store who confirmed it was functioning correctly (showed the correct amount of CCA under test).
I suspect a short, but that does not explain how the battery is recharging without running. Also, I have seen it go from functioning to non-functioning without moving, so I am doubtful about a loose wire. Does the hybrid battery charge the 12V while the car is off?

 

Measured at the battery. The weirdness continues, however. I removed the cable from the negative battery terminal and waited a day. The battery came up to over 12V. I don't see how this is possible since there is no completed circuit. I am also now fully convinced that the car is going into accessory mode at some point while the car is parked which is why the battery is being drained. I also think that at some point, the car after discharging the battery resets and drops out of accessory mode and the hybrid battery recharges the 12V while the car is sitting there. I did have the control board changed out do to the recall about 10 years ago when the dash went dark, but it has been staple up until now....