Is there a part of this site I haven't found, or perhaps another site somewhere, that shows a person's profile? Kinda like WoW Armory or something similar, that lets me look someone up, see who they play, how they've been doing, maybe even gear used in recent games?

You can look up a summoner profile ingame. You can also use lolbase.com but it only shows stats from uploaded matches.

For matches you can use the PVP.net (lol's client that is used to find matches) and it says your games you played and top three used champs.

You can also use a site called lolbase that you can use to upload your logs and get more detailed information on yourself and other players.

http://www.lolbase.net/downloads

You just download the client and point it to League of Legends\air\logs