Do I have to do that at the immigration where I live in Thailand or can I do that at any immigration in Thailand or even at my home country, what I rather doubt?

Suppose I would leave the country before the visa will be expired without a reentry visa to change the date of the yearly extension, how can I get a new marriage visa?

There are no "marriage" Visas although "O" visas based on being married, single or multiple entry are obtainable from Thai embassy's / Consulates.

An "extension of stay" (not visa) can only be obtained from a Thai immigration Office.

If you want to apply for a one year extension of stay based upon marriage you have to apply for it at the immigration office where you live.

If you left without a re-entry permit you would need to get a new non-o visa and apply for the extension during the last 30 days of the 90 entry the visa gives.

thank you ubonjoe, so no change to make that in BKK, if I life on Samui?

New one: you mean, if I left without a reentry permit...if I would leave and come back with a reentry permit the old expire date would still be valid.

You cannot apply for an extension in Bangkok if you live in Samui. The only thing they will do for non Bangkok residents is a change of visa status if you don't already have a non-o visa.

I didn't see I had forgotten the out. It is there now.

I have a 1 year working permit too, which is extended year by year. So, it seems, that that excludes leaving the country without a reentry permit, and it means that you are stick to that date of the yearly 1 month period for the extension of the marriage visa. Or is there any possibility to change that to another date?

Your work permit is not tied to your permit to stay date. Nor is your extension of stay based upon marriage tied to it. You could go out and get a non-o and apply for a new extension of stay without doing anything with your work permit.

If you wanted to change the date of your work permit renewal you would have to cancel your work permit and apply for a new one.