Aside from diplomatic victory (introduced in Diplomacy) the only other way to win is to defeat all enemies.  An enemy is considered defeated when they lose all their planets, but most players (including the AI) will surrender when they've been beat up to the point of no recovery.

Back in Entrenchment and Vanilla, it was possible to make a 5 person team in a 6-way FFA and have all five people win, which was kinda cheesy.  Allied victory limits the number of people who can win as a team, meaning that eventually that massive team is going to have to backstab each other.

I can't speak for the Diplomacy category, but the other three categories are somewhat flawed.  Often times when I'm putting together a 15-minute mark rush against an unfair AI, I'm #2 for virtually every category even as I'm kicking its ass.  While these categories generally can be used to determine where a player stands, a good tactical position can prevail even while losing on virtually every category.

Colonization is number of planets owned.  Culture is the % of phase lanes in the galaxy under the effect of your culture.  Research is the number of technologies researched (a high tech research is considered equal to a low tech research, clearly leaving this a useless measurement).  Fleet merely counts the size of your fleet.  The quality of the units making up your fleet doesn't count.  Capital ships are a composite of how many caps you own and what level they are; I'm not sure the exact weighting.  Defenses is a composite of how many tactical defense structures and starbases you own; I know starbases are heavily weighted in this metric, so it's basically a measure of who owns the most starbases.  Economy, as you say, is obvious.

No, you have to scout and figure that out for yourself.  It's usually fairly obvious if you're actively snooping around.