Sounds like you have two different problems.
1. Not going to the first barrel
2. Not stopping after the last barrel (after finish line)
I have seen some horses that get "barrel headed" and do not stop after the run. Some people don't fix this. I personally think it is dangerous and must be fixed. I had a horse that became very "up" in a competitive situation and I do not like the idea of galloping at a fence, not sure if he will slow/stop/turn in time. :)
You say you have tried walking and troting through the pattern. Do you still have both problems when he does that?
Have you worked on rating your horse? You start this by walking toward the barrel. When the horse's head is even with the barrel you stop (whoa) and back up a step. Do the pattern at a walk like this.. do it MANY times. This teaches the horse to rate (slow down and adjust speed) coming into the turn. This may also help your horse to listen to you better rather than running off on his own.
There are some trainer who believe you should not run patterns at speed during the practice. Only at shows. Something to consider.
Do you always run the same pattern? Try different patterns. You have to mix things up. If the horse always knows what to expect, he will go off on his own. You want him not 100% on what it coming next - you want him to listen and look to you for direction.
What do you do when the horse misses the first barrel? Do you let him finish the pattern? If so, you are reinforcing the wrong behaviour. If you point to the 1st barrel and he runs off to the second, you need to correct him. STOP the pattern, turn him in circles (back him up, etc... make him WORK). Then calmly walk back to the beginning and start over. Keep correcting until he listens and gets is right. THEN praise.
What about not stopping at the end? I wouldn't let him get to the finish line if he wasn't listening. Turn him 180 and do the last barrel again.... and again and again and again. Don't stop until you have control back on the rundown. YOU have to pick the speed, not the horse.
Practice the final dash at different speeds. Come out of the last barrel and break it down to a trot. Trot across the finish line.
With a horse behaving like you describe, I would not go above a trot until you have walk and trot at 100%. That means MANY repetitions with stops, gait changes and other "surprises" thrown in to add variety and make sure the horse was listening to ME - not just going through the motions. Head down, loose rein, soft and responsive, not fighting to go faster. At that point, you can try a controlled lope and work from there.