I have an open fuel cap (Check Fuel Cap), but the cap is tightly screwed on?

It looks like you have fixed the cranking problem. thanks for the video, it was good. Please check the O ring seal on the gas cap, it must be in good shape. If so, you may have an EVAP vent line that has come off or the hose is no good.

Check Fuel Cap Indicator

Note: The check fuel cap indicator on the Hanger is a dedicated output signal that is controlled by the PCM.

The check fuel cap indicator is a communications network message sent by the PCM. The PCM sends the message to illuminate the lamp when the strategy determines that there is a failure in the vapor management system due to the fuel filler cap not being sealed correctly. This would be detected by the inability to pull vacuum in the fuel tank, after a fueling event.

Check out the images (below). Please let us know how it goes

Now I remember that I recently left my car with a mechanic to change some parts, including the fuel filter. Unfortunately, he couldn't do it because he didn't have a tool in this case (that is, he tried but didn't succeed) and I think that he didn't squeak the hoses well again and that's the reason.
https://www.ebay.com/itm/364340969337

Regarding the problem with the ignition when the car is in parking lot (P), I haven't fixed it just with this video, I want to know if I should try this way, that is, if it will have an effect or if I should change the cable.

Regarding the 0 ring on the tank cap, I don't see any splits or tears, but it seems stiff and shrunk to me. I don't know if this is normal, but it also seems like it unscrews easily and doesn't pop off. Does this cap have to be replaced at a certain mileage? If yes, then it's time to change it.

It doesn't have a particular mileage, but if the part has failed it is time for a replacement. Sorry, we only handle one problem per thread, please post your new question here, you must be logged in.

https://www.2carpros.com/questions/new

Cheers, Ken

Yesterday I replaced the fuel cap with a new one.
https://www.ebay.com/itm/265918823337

but it turned out that the problem is not in it. After starting the engine, the display continues to say (Check Fuel Cap). I am not clear from here on the father which part to check and replace. You mention a disconnected breather or a bad hose but where to put them I'm looking to check? It made me think that above the spare tire next to the beam, when I squatted down next to the rear tire (regardless of which of the two) and looked ahead, I saw a socket that still seems to me to be not fastened well. And yet I don't know if This is the way it should be.

https://www.youtube.com/watch?v=dBT7GmaZbCA
https://www.youtube.com/watch?v=cU1vHBgLmLA&t=21s
https://www.youtube.com/watch?v=BwuEjw-iXI8
I watched these videos but I can't figure out apart from cleaning, which part to check and do the battery terminals hang after I changed the cap to clear this error. I'm not sure what to do.

Thanks for the images. it is difficult to tell what you are talking about, but I can tell you the EVAP system can have no leaks. So, if you see a hose not seated, I would reattach it and use the video above to clear the codes.

So, I found the problem, a broken hose for the vapors, I replaced it, but in order to clear this error from the car's computer (Check Fuel Cap), the battery terminals must be disconnected for at least ten minutes.

Glad you could get it fixed, thanks for letting us know. Please use 2CarPros anytime we are here to help.