1. When Zeus arrived he was very pushy and in my space all the time. He very rarely tries that with me anymore, but he will test you. If you allow it even once, he will very quickly redevelop that habit of pushing into you. Watch for little things, like him leaning his shoulder toward you (or even taking a tiny step toward you) when you are grooming him, such that you feel like you need to step back from him. DO NOT step back! Use your thumb to apply direct pressure to his shoulder or use driving pressure to ask him to stop leaning or stepping into your bubble. If you move your feet away from him when he is trying to apply pressure to you, you have just taught him that he is in charge of you. Lot's of backing can help with this also. Try backing him to where you want to go instead of leading him.
2. Lots of Circles and Figure 8's. I don't like to hold a horse on a circle. I set him on the circle and it is his job to stay on that circle at the given pace until I tell him differently. Zeus is still trying to figure out how to stay on the circle. If he meanders off, that's not a problem, I just use a direct rein to get him back on the circle. As soon as he is back on your line, go back to a totally loose rein. Stay calm and repeat as necessary. It will take a LOT of repetition--but he WILL get it and in the end you will have a much more pleasant riding experience that if you are always having to hold him to your line. This is an important "grade school" exercise that he needs to excel at before you move on to teaching him collection.
When he "gets" this at a trot, do it at a lope. Currently, he is more comfortable circling to the right than he is to the left. Try going to the left twice as much as to the right until he gets more comfortable. Also, sometimes directional things like this are simply due to the horse not being totally comfortable being ridden yet. Try doing #3 below.
3. Loping. We've mainly only done a little bit of this in the barn. I've done a few straight lines across the pasture, but the mud is still pretty slippery, so I haven't tried to circle him outside at a lope. He has a hard enough time keeping his feet under him at a trot, sometimes! He still needs a lot of lope time.
I recommend finding a corral or arena with good footing and just asking him to lope for 10-15 minutes straight. If he slows down, no problem, just ask him to go again. You may need a spanker to over-under him so he knows you are serious. I start by gently swinging the spanker across my front and work my way back to his hindquarters, increasing in intensity as needed. Allow him to rest for 5-10 minutes in a spot where he didn't want to go before (not at the gate!) and then repeat.
For this exercise, you are just trying to get him comfortable loping and to teach him to maintain the gait you asked for until you ask for something different. Don't worry about steering him. Keep the reins loose. If he goes into a corner, just keep asking for the lope. Let him figure out how to get himself out of the corner. As long as he is loping, he gets no additional pressure from you.
This exercise can be done at all gaits, so if you don't feel ready to try him at a lope right away, practice this for 15-20 minutes at a trot. Rest for several minutes and repeat.
Thanks for allowing me to play with your horse. I enjoy him. Call or email me with any questions. Good Luck!