There are Major Powers and minor factions. The major powers are the Empire, Federation, and Alliance. There are tens of thousands of minor factions within these powers. Major powers control entire systems and minor factions can control individual stations and places within a system, sometimes all of them.

Gain rep with a minor faction at a station to get better missions there and better pay. Also turns them on the HUD green if they are your buddies.

Major powers control systems, marked by coloring them in the galaxy map. Red for Feds, blue for Imperials, green for Alliance. It's incredibly difficult to change major power control, but minor factions can swap more easily.

A bit, but it's difficult or impossible for a solo pilot to do much, but it can happen. Groups make it easier.

You can't join factions like the major powers or any minor factions. There is powerplay, where you pledge to support a political figurehead and their ideas, but you never actually join a faction. This isn't a good game area to try as a new player, and in general it's a poorly made and now often ignored area of the game.

You could if you wanted. Some have been exterminated. (minor factions)

Cops are a part of whatever major power owns the system. Federal security in Fed systems, Imperials in Empire systems, etc. Green means they are friendly with you due to rep.

Minor factions might fight due to a huge gap in system influence, spawning CZs. They can last 3-28 days if I recall. They are not for bounty hunting as all targets are clean, and all are high grade military targets fighting for their minor faction.