Ok,
Here are some tips regarding reading codes and all that jazz.
Number one rule....always start with the lowest numbered code first!
So in your case, test/check or replace the canister purge solenoid. It is the lowest numbered code in your list, and as I've seen in the past can cause other codes not to clear. Is it directly related to your fuel cut issue.....I don't know. But it's good troubleshooting practice to start with the items you know to be a problem and then move onto the unknowns. So if you know or can verify by testing that the canister purge solenoid is bad, replace that, reset the ECU, and check the codes again.
As for connectors.....as it's been mentioned, the black connectors are only for reading stored codes in the ECU's memory. These could be codes that are still present OR they could be codes from months or even years ago that are still stored in the ECU's memory. It could also contain erroneous codes that get stored for no reason at all. So you need to use some judgment and additional checks before assuming all those parts are bad. Typically I would suggest starting with the black connectors which pull stored codes (this is also called the read mode check).
After you've done the stored code check, I'd suggest using the green connectors to check for active codes in the ECU that are currently throwing faults. This mode is also called d-check mode. This mode will typically ONLY display one code. It will be the lowest numbered code. For example if your knock sensor, canister purge solenoid, and wastegate solenoid are all bad, the d-check mode will only typically display the code for the knock sensor. The proper troubleshooting procedure would be to verify the knock sensor is bad, replace it and perform the d-check mode again to see if there are any additional active codes in the ECU. You would keep repeating this process until you no longer get any more active codes.
To clear the stored codes from the ECU's memory, you would use BOTH the black and green diagnostic connectors. This should clear stored codes from the ECU's memory. I have run across times where people can not get the codes to clear with the battery dance method or by removing the ECU/TCU's backup power supply fuse. So this should be a sure-fire way to clear any stored codes.
Regarding the Power light, if the TCU detects a fault in the transmission it will blink the power light 16 times on start up. This is NOT an indicator of any particular code, but rather that there is a fault detected in the transmission somewhere and that a diagnostic check should be performed. Like with the ECU, my recommendation is to start with the stored code retrieval procedure and then move to the active code retrieval. (I need to reword the
instructions on my site, as the FSM wording is a little confusing).