Heh, I know what you're talking about.  Just yesterday I played a tight, high-level game (3v3) as Sedna, and I was really struggling with watching my teammates' health, watching my opponents' health, etc. esp. when we all got into 3v3 scrums.  I know I f'ed up a couple times with my heals because I couldn't track peoples' stats fast enough.

A couple things:

1.  Use your scroll-wheel to zoom in when things get crazy, like a 3v3 scrum.  In general, it's best to play maps zoomed out as far as you can, so that you can get a perspective on what's going on everywhere (do I need to port over to help out/protect a tower, cap a flag, etc.) but in tight battles, zooming in can be really helpful.  I esp. find myself zoomed in a lot on Prison, where things are pretty much always a 3v3 scrum.

2.  The more you play, the easier it will get.  You'll learn the sound-cues associated with all the skills, you'll be able to pick out an Oozing beast from a mile away, etc.

When playing against Rook, be wary of enemy Dg's that port in right undernearth him (to a tower).  I find those the hardest to see, but you gotta look for that swirl.

It can be easy to get lost in the fray sometimes.. but.. generally:

Play zoomed out, but not to the point you can't see animations (like a teleport).  Easy to test this point.. just grab a tp and zoom out as far as you can go.. start casting the tp and if you can still see it, great.. if not, zoom in a little more.  Probably don't want to finish the tp until you get it sorted out tho.  Anyway.. this let's you still see enemy abilities as well as see most of the map.  A great mix of both worlds.

Oak and Rook are the only ones I can get lost in.. but.. through maneuvering of the camera.. I can usually spot my target, and I know all the animations / how much damage they do to see enemy abilities

Yup! Certain non-game-changing mods will allow you to join games. Things that would change gameplay like the Favor Mod would bar you from entering a game, whereas the godlike team panel will let you join games even if the other players do not have it installed or enabled.

ya the UI mods are AMAZING.  Srsly.. I don't remember how I played w/out them lol.  So handy to see your teammates mana, what items they have if you want to give suggestions, favor, gold to plan citadel upgrades, your own hp / mana numbers stickied so you can get technical in your calculations on living or dying in battle.. mmmm

UI Mods to get:

http://forums.demigodthegame.com/369937

http://forums.demigodthegame.com/369526

http://forums.demigodthegame.com/369556

Follow install directions if necessary.. I thiiiink they are all .exe and autoinstall when you run after downloading.. but if not, directions are in the threads

Predict whats going to happen before it actually does, no seriously plan your move ahead of time and put every possible situation into play really.

There's already some good advice in this thread.  I completely agree with getting a UI mod that let's you see what your allies have.  Also, get a good zoom level that you are comfortable with and don't be afraid to rotate the camera.  

Most of the time, these kinds of traffic jams happen around valuable flags and portals in the mid to late game.  When you lose sight of your demigod, it's time to move out of the fray.  Probably more players get ganked and killed in these situations than any other.  Since traffic jams are usually around flags, make sure that your team controls the flag or that it's neutral (yellow)--otherwise, you are asking to get ganked by a teleporting UB or some other ganker.  Sometimes you have to fight in a mob, such as when you are trying to save your portal from being captured.  Still, when you lose sight of your demigod, you can almost assume that he/she will be dead shortly.