How do you read the radar in multiplayer?
For Shapes:
Large orange circle: An enemy pilot
Small orange circle: Enemy AI controlled entity (other than titan)
Orange arrow: An enemy titan
The rest of your questions:
A whole 1/8th of the circle can be colored in the enemy color (orange). What does this mean?
This is if there are shots being fired within a near area by an enemy.
Sometimes, a faded orange circle will appear around a section of the map, which may or may not have anything to do with map objectives. What's this for?
In my experience this is where enemy spawns are happening for the current time. The enemy can see this too. Played a game of Pilot vs Pilot last night and the enemies only showed up there.
Lastly, sometimes the outer ring of the radar has solid colored in orange sections. What do these indicate?
This is similar to the shots being fired but further away from your current location.
Are there any other indicators on the radar that I have forgotten about or haven't seen yet?
I'm not sure what you mean by diamonds. So I haven't answered that. Other than that I think you got them all.