You are largely wrong. British law takes the idea that “a gentleman’s word is his bond” seriously. Even more so in the 17th century. Of course children, the insane, and women could not witness legal documents, but no special official was required (or existed). I’ve learned quite a bit about wills and probate recently, dealing with my father’s estate. The UK isn’t the 51st state, it’s another country, we do things differently here.
I'm not from the US. I'm a legal academic in Australia. I teach, and have published on, Australian private law which (as I'm sure you appreciate) incorporates significant elements of English law in virtue of the general rules of reception of English law in a "settled" colony.
Unless you're an English lawyer, I'm confident that I know more about English law than you, have read more cases on wills and the transfer of land than you, and have a better grasp of English legal history than you.
Yes he does. He tells us there is NONE. You might find that implausible, but that is what the author explicitly states.
JRRT does not say that there are no public officials in the shire. To the contrary, he tells us that there are some.
How is Sam Gamgee appointed Mayor? The Appendices refer to him being "elected". By whom? And by what process? Certainly not secret ballot, which was invented in Victoria in the mid-19th century and would not have existed in The Shire. By the Shirrifs wandering The Shire and taking the temperature at public meetings? By a public meeting at Hobbiton? By some other method? We can be confident that some method existed, but we don't know what it was.
By the way, Googling Sam Gamgee elected Mayor turned up this:
https://tolkiengateway.net/wiki/Samwise_Gamgee
And from that website I learned that, according to a letter that JRRT wrote, "Soon after his election he [Samwise] established a rule of succession and inheritance in situations such as the headless Baggins family, since Bilbo and Frodo went over the sea and was impossible to presume death." So not only did JRRT recognise that The Shire had a system of private law (which is already obvious from what we are told in the stories), in his letters he was even making up details about how the rules of private law might be changed - namely, by Mayoral decision.
I regard this as consistent with my view that details of the government of a fantasy world can be made up, as needed, based on a combination of established fiction, common sense, and what seems like it would be fun or interesting, without needing to specify it in advance.