You need to ask your local immigration office for their policy.
If you mean under consideration period (3-4 weeks) for your 1-year extension based on marriage to a Thai, etc, the answer is no, Immigration offices don't hold your passport for the whole u/c period. But there are some offices that keep your passport for up to a few days when you apply for your extension.
No, nothing can be done at the airport for switching your passports either on departure or arrival. You have to go to the immigration office anyway to transfer stamps, sooner or later.
1. The best is to go to the local immigration office that issued your extension and re-entry permit to transfer stamps to the new passport before you travel. Then you don't have to bring both passports around.
2. Alternatively, you go travel with both passports, showing your old passport to get the departure and entry stamps on your new passport at passport control on the Thailand side. Then go to the immigration office to apply for transfer stamps properly after you are back to Thailand. (Not sure if you would be asked at passport control/immigration office why you haven't transferred stamps before you leave or not.)
In this case, make sure you are stamped until the same date as your extension with Non-RE category on your arrival to Thailand, not a 30-day stamp, etc.
The rule is clear. Visa-exempt entries through land border checkpoints are allowed two times per calendar year. They can enter Thailand on visa-exempt entry by land one more time this year.
What we can advise is to avoid border-bounce at the Aranyaprathet / Poipet border with those histories.
Almost yes, you have to apply for an initial Non-O visa based on marriage, then apply for a 1-year extension for the same reason at the local immigration office.
You can apply for an initial Non-O visa at one of the following: the Thai embassy in your home country, the Thai embassy in a neighbouring country, or the local immigration office (except a few) in your province in Thailand.
Detailed requirements are different by location. Decide where you want to apply for your Non-O visa. Then you need to check their website (or contact them directly) for the requirements.