Generally speaking the "rating structure" is a lattice where you can, for any two players A and B, tell whether A is a better player than B or the other way around. Elo, Glicko, etc. are embeddings of this lattice on the real line (much like the utility functions of microeconomics are real embeddings of preference lattices).
Not really. You tend to have cycles, where person A can beat person B who beats person C who beats person A.
There's a guy who was even with the guys playing Go who were 2 or 3 stones weaker than me that would tend to beat me because of some of the unorthodox things he did. (Eventually I strengthened my game against these things).
Considering ratings to be a total ordering is a useful approximation.
Others have pointed out how there is a psychological aspect of rating systems, and no developer wants to constantly field complaints. That said, I believe the answer is yes. A rating system derives meaningfulness from its predictive power. In other words, people want to know how good they actually are compared to one another.
There's an interesting question about how you rate players who use fundamentally different strategies. For instance in RTS games, should you match boom vs blitz players of otherwise equivalent elo? Or should you instead (try to) construct a classifier to determine which type of player someone is, and then have a rank against each other type of player and match them according to that rank?
That's why people use barcode names (llll1l1l1l1l111l) in starcraft :)