Yes, you are entitled to enter 2 times using visa exempt if your from a qualifying country and have the supporting criteria if asked without any issue.
You can enter using exempt more than your entitlement but it becomes a matter for the passport officers discretion based on your travel history using exempt. (Eg you spend less time abroad than in Thailand, so ok).
2 maxed out exempt entries with maxed out extensions is going to equal 180 days in the future, so this will lead to a lot more conversations at passport control for 3rd attempts.
i think false is a bit strong, especially as it does appear on nearly all consulate/embassy web sites. I believe what it is trying to say is people from exempt countries are 'entitled' to a maximum of 2 entries per year (as long as they meet the other criteria if asked for), and above 2 it becomes discretionary on the immigration officer based on your history etc. Previously land and sea immigration officers werent allowed this discretion.
Good question, been wondering about that myself. I suppose local immigration could put a stamp with the visa type, but this generally only reflects your permision to stay date, not visa expiration date.
You will have to show the visa which if e-visa is a pdf, and wont be any type of stamp in your passport. (Only your previous entry and exit stamps). It is highly unlikely any other documentation is required at that point, unless they specifically target soft option as the other options havent got simple suporting paperwork that can be shown at passport control.
for extensions i agree, its usual process to produce the same documentation. But i very much doubt that anything will be checked on entry by the boarder guard on a visit. While the soft options could require a pdf with bookings/schedule as it would be a quick check, non soft options would be difficult to have a document that could be quickly checked.
Assuming your enquiring about an e-visa DTV, you will not have a physical visa or visa stamp in your passport to transfer to my knowledge, depending on your usage you may have entry and exit stamps. Most instances on change of passport, people go to imigration to transfer Extension of stay and re-entry permits not visa's. So there are two scenarios in that you are entering Thailand and you new passport doesnt match your pdf visa, or your in Thailand and change passport, the 2nd one you can probably do something at local immigration, but the first one needs understanding before you travel.
Are they not asking about the Kor Ror 22 which should have been obtained from a local Amphur if marriage was abroad and involved a Thai citizen, or Kor Ror 2 if married in Thailand? Either should just involve a trip to the Amphur to produce a new copy.