Cow work- right on schedule

Last Sunday a neighbor invited us to come work a few cows with our horses. I decided to ride Chief for the initial rounding-up of the herd, and then threw my saddle onto Tex to see if he would give cutting in the small pen a try. (by the way, he has been around in a pen with cows about two other times, and has been tied to the fence near them during ropings and team sorting multiple times.)
In the pen with about 10 cow-cal pairs, I started by asking him to move one light-colored calf around the pen. He wasn't focusing at first, but the longer we worked, the more he seemed to catch on. It also helped me understand why it's important fo him to have a strong stop, fast go, and fast turn with a light front-end.

By the end of our little 30 minute work time he had sorted all of the calves away from the cows. They didn't stay in the corner where we put them, but that's ok. If we get the chance again to sort cows for real, I'll feel comfortable getting on him and using him for that work.

Pros- 1. There is definitely some interest in cows, and a little bit of a natural cutting feel to him.
2. Tex is being very calm in the arena.

Cons/things to work on-
1. I can no longer allow him to be lazy with his turns and going forward.
2. I must get on and off him multiple times to work on standing still. He is starting to walk off when I mount up, even though I've always refrained from moving for at least 20-30 seconds after I mount up.
3. He is still very nervous outside the arena. We nearly had a wreck once out on the trail with 3 other friends.
4. On the trail, he is pushing through the bit, wanting to go, go, go. But when we do pick up a trot, he gets super nervous. I'm not sure what to do about that. Suggestions?

Another random
TM