"a dear person"
No, I believe друг and дорогой are completely unrelated. "Друг" has a very old indo-european origin and has relatives in languages as remote as Icelandic, at least according to this page: https://ru.wiktionary.org/wiki/%D0%B4%D1%80%D1%83%D0%B3
As for "дорогой" meaning both "dear" and "expensive", I don't think Russian is that special: I believe English "dear" used to mean "expensive" as well, which is still manifest in expressions like "I paid dearly".
Yes, I don't know about "used to", but in UK English, "dear" can certainly still mean: "expensive". That was what prompted my question above (to which you already replied) about which sense is meant here. It seems that in Russian, as in English, it just means: "valuable", or "valued" - which could be in purely monetary or sentimental terms - or occasionally both. It's possible to have a possession which is both precious to you AND costly; an example would be my grandmother's wedding ring. That is "dear" in both senses.
That's true - very much out-of-fashion these days, and might have been, even without Tolkien, but the same underlying idea, and not a jibe about anyone's spending habits. Although, to confuse things further, "precious", as a description of someone, can be derogatory, meaning they are rather pretentious or exaggeratedly demonstrative. I've not heard that one much lately, either.