Ok for your second question, biggest thing is I know how tempting it is with them to keep pestering to keep them going. But your pestering all the time is not going to help the situation. You need to install 'gears' in him. Again it's another pretty typical ottb trait, they are unsure of what they need to do, so they'll switch gaits. My boy used to do it all the time as well so I know how frustrating it is!
So, get him into trot, and then stop fidgeting around, just let him trot. The second he slows to come back to walk/halt, put your leg on again. Then when he goes, just let him, don't touch him at all, let your reins out, don't pester with your legs, just let him go. And when he slows, ask again. He'll come to the realisation that he just has to keep at the same pace until you ask. Keep him trotting until he'll trot consistently on his own for one length of the arena, then halfway down the next long side ask for him to walk. If he halts, pick him back up to trot again straight away, until he's moving by himself then ask for walk again. Don't pull back on your reins for walk, just look up, take your legs off and lift your chest. By the sounds of him I'd be confident that he will react to just that for a downward!