First of all WELCOME to our board!!

In response to your question, it seems as though we can be guilty of working on our horses more than we work on ourselves. But, before I go any further what type and size saddle are you using? does it seem to fit you well or is it too big?

What is your hand position heading into and through that second barrel?

Many Many things can cause the balance to be off.  Give us as much info about your riding style and your horses running style (turns more on forehand, 4-wheel drive, free-running ect) as you can so we can get a better mental picture of what may be going wrong 

Sounds to me like you just need more time in the saddle to develop those good barrel racing muscles.  Flopping around like a rag doll is all part of learning especially if your mount already knows the drill and you are new. 

I would work on circles and rollbacks and doing slow pattern work until you feel comfortable. If you can't get your rein back you've added too much speed too quick.  Get the basics down and progress slowly.  It is the safest way to learn. 

He may know his stuff but you need to have a good seat in order to be a compliment to him.  Take your time, it will be well worth the wait.

One thing that you need to realize is that you haven't worked this horse on barrels much.  So of course, it will take time to get used to it.  As for your second barrel problem, one thing that I always like to do is "drop down like a cutter".  Meaning when you reach your point of rate sit down on your pockets ("melt in the saddle").  Also, pick up your inside hip just slightly and you'll realize that automatically you are tightening your whole outside leg, causing your horse to follow his pocket instead of going wide.  If you feel like you are getting thrown out of the saddle, when you "drope down like a cutter", you need to push against your saddle horn, don't just hold on to it.  If you push against the saddle horn, it will keep you in the saddle and more secure.   I would also try lengthing your stirrups a noche and see how it goes. You may have too short of stirrups.  Try it and see if that helps. 

Best of luck and I hope this helped give you more ideas!

Try turning and looking for the third barrel.  Look where you are wanting to go not where you are at the moment.  Looking for the next barrel does amazing things for your body language.

A good trick to help you get your outside rein back is "touch and slide".  Using your free hand, touch the hand holding the rein...now slide the free hand down the rein to the spot you  want to pick up.  Does that make sense?