You do not have a Visa extension. Forget the word Visa, that is not what you have.
You have an extension of stay. Different laws different ministerium.
A Visa you buy on a consulate or embassy outside thailand. You cannot extend this. Only buy a new outside thailand.
An extension of stay is an extension of the stamp in your passport that says Admitted to. Not an extension of the Visa.
Your extension of stay is not valid after the day you stop working.
Simply put, an extension of stay does not give you right to stay in thailand, if the base for it does not exist. No work permit, then no extension of stay. Date on extension of stay is irrelevant. Date of the work permit IS relevant.
If you hesitate too long before you solve this situation, you will have even bigger problem. If caught working without a working permit is not good. And on top of that is here illegally.. Think. Act.
If caught by police before you leave Thailand, depending of how long overstay it is at that time, beside fines you are facing a situation you no longer can return to Thailand for some years.
You really need to understand the difference between a Visa and an extension of stay. According to your photo, you do not have a visa. Maybe you once had a visa, but that is since long used and invalid. You have an extension of stay as you earlier applied for. An extension of stay based on a work permit is not a visa,
Your extension of stay is dependent of your work permit and is not valid for stay in Thailand after your work permit ended. Hence you are now on overstay, you are in Thailand illegally. This is a serious situation.
The only solution you have is to leave Thailand, to buy a new visa, a Non B visa. Then with this Visa you can return and start the process again with applying for a work permit.
And later based on this apply for a new extension of stay.
It has nothing to do with where you usually live, as the registration for change of address you do in Europe. They want to know where you currently stay every day. It is similar to the electronic tag. They claim it is to prevent terrorism.
Not new. The Hotel report you as staying there, hence you need to file a new TM30 when you arrive back. I was forced to do it in Prachin Buri the last time I stayed one night in a Bangkok hotel.
The requirement is there to be able track where you are all time. Not only where you arrived initially.