I don't have specific numbers on the relays and exit nodes/relays the FBI is running, but it's certainly in the minority. Basically the only way people using Tor for illegal activities are caught is because they gave themselves away -- typically by revealing information over Tor that could be linked to their identity (like using the same username over multiple sites).

It might be helpful to think of the potential risks of using Tor as sitting outside FBI Headquarters and connecting to a network called "Free Wi-Fi" with an anonymous laptop. You can't be sure the FBI is running it, but there's a chance. Ignoring physical surveillance for a moment, "they" can't tell who you are unless you give them that information. If you use a browser extension like HTTPS Everywhere to encrypt your web traffic on sites that support HTTPS, you can be protected from snooping. This is an imperfect analogy, but you get the point.

Regarding your other question, an exit relay is just what I've been calling an exit node. It's the last stop before your traffic leaves the Tor network, and to the rest of the internet, the exit relay you're using is where it looks like your traffic is coming from. A non-exit relay is one of the intermediate steps before your traffic reaches the exit node/relay. The Tor overview and FAQ have tons more information, if you're curious.