I tend to think of it as a sport. If you're just playing with friends or random people you come across, who cares if you win or lose? It's there for fun. Elo is just what they called the rating system they use, even though it's not a true elo system (google it if you're really curious, and for sanity sake don't just take 1 source as the truth.).
Personally I don't consider high elo to give bragging rights, if anything it flags you as a reason to complain about Riot's matchmaking. For instance, last night I was paired against a 2200 ranked elo player in lane and lost miserably. My normal elo sits around 1600 and my ranked solo is 1337 (Yes I stopped there on purpose). Now, I'm willing to argue that I should be around 1800 if I really cared to do ranked, but that's still a far cry from the 2200 I fought and got crushed by. I didn't really care, since I don't take the game too seriously, but my teammates used his elo as an argument for why I lost my lane and argued it against Riot's matchmaking.
Competitive play on the other hands has always been a multi billion dollar industry, and as such is taken beyond that of a "just for fun" thing.
The rest of your questions were more or less answered, I just felt like giving my opinion on the matter.