I agree that "thank you" is slightly more formal than "thanks". I suspect if I was addressing the Queen I'd say "Thank you, ma'am". I tend to use "thank you" for emphasis, to show that I am really grateful, and it's not just a polite formality. Otherwise there's not a great difference, in my opinion of common usage. (Except for 'giving thanks', but that's a whole other kettle of fish.)
"According to so-and-so" is a well-accepted phrase to indicate a certain level of authority. It doesn't mean authority in the sense of indisputable, world-class, absolute authority. It means authority only at the level of indicating that this is the person to whom the statement can be attributed.
It therefore has rather wide usage, all the way from according to my brother, Nancy went to the dance with Frank, right on up to according to Einstein, the universe is curved.
The lowest level of its reference to authority, however, is the second person, according to you, which sometimes is used sarcastically, thus: According to you, I should never have bought stock in Microsoft. Boy, were you wrong.
We don't take it down to the first person, because that becomes overly emphatic, or to put it another way, redundant. According to me is simply saying, this is what I say. It isn't considered necessary to say this is what I say when you say something. You just say it, don't you? I can tell you Nancy went to the dance with Frank without saying According to me, Nancy went to the dance with Frank, you see?
If we look at your examples, Mari-Lou, we can see a little bit more of why the usage sounds a bit strange to native English speakers. Since we do imbue a bit of authority to the according to locution, we usually feel according to my brother, our father will go to a nursing home someday does sound odd; we would instead tend to say my brother thinks our father will go someday, and the response would be but I don't think so rather than according to me he won't. According to in standard English usage is given some weight of authority, even at the lowest level. It isn't taken as purely opinion.
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One problem is that the term idiom has various definitions. I would be surprised if the websites you refer to make the claim that a.) according to and at first glance are idioms and b.) such idioms are not used in formal conversations or writing. This is simply not true.
A narrower definition of idiom is that an idiom is a phrase whose meaning cannot be deduced from its constituent words. Examples are:
It is true that informal language abounds with idioms in this sense of the word. It is also true that you would not expect to see some of these idioms in formal conversations or writing. It is not true, however, to claim that no idioms are found in such situations.
Here are just three examples from the Telegraph, a serious British newspaper: