ATC light on, "VSC off" light flashing, CEL on... fault code is P0037 - the O2 sensor? What?

2010 Corolla with the 1.8L. I posted last year about going through a deep puddle and seeing this issue, but it went away on its own. My child had this car in college this past year. Just got it back, and the lights are on again. I used my scan tool and only this one P0037 code is being thrown. I cleared it, but the lights came back on after 1/2 mile of driving.

I'm guessing there might be water-caused corrosion on the connector for the 02 sensor, but the bigger question is why would this trigger those other lights? The car drives normally. I read about VSC reducing power to the wheel(s) it senses are spinning, but I detect nothing. As I said, no issues driving at all. Any clues or troubleshooting tips appreciated! The car is up for inspection and it will not pass with the CEL lit! Thanks.
Here's something... I just put the car on ramps and took a look at the wire from the downstream O2 sensor. The wires themselves look intact but the insulation has some damage. I'm afraid to try disconnecting it as I usually wind up making things worse, but I'll ask: do any of these connections come apart easily? Thanks again!

More researching... this is the correct o2 sensor (Denso 234-4305). I see it has a clip-on connector. Is that inside the rubber "boot" in the foreground of my picture above? I don't believe I saw it when I was under the car (though I didn't get a great look at the o2 sensor itself). Thanks once again for any helpful tips!

Sorry for cluttering up the thread. After watching a few more u-tubes I see this clip-on connector is inside the passenger compartment, after the rubber boot. And, the boot is included with a new sensor. So, hopefully my last question here :) For a very occasional weekend mechanic guy, how hard a job is this? I'm pretty sure I have the special split socket for the sensor, but just a basic ratchet set and breaker bar. I can see trouble ahead trying to get the old sensor out! Anybody with experience want to share? Of course this might not fix my original issue, but if the sensor is bad and I have a shot at clearing the CEL, it needs to be done! Thanks.