I hope you are asking how to do all this stuff from the ground and not while riding! Haha! Otherwise I'm wasting my time...
How do you tell your horse that he has to cross legs and step to the side?
It helps to do this up against one of the arena walls. Have your horse standing perpendicular to the fence with his nose to the wall and his hindquarters to the inside of the arena. It helps to have a dressage whip or some other sort of "handy stick." For me, I start by putting the hand with the lead rope in it up by the horse's face. Start "tapping" the air. If the horse moves his front end over, release the pressure. If he doesn't, you can keep trying or bump him on the nose. If he reacts drastically, keep him on a short line and bring him back up to the fence. Start again with the gentlest cue. To move the hindquarters, use your stick and tap the air again. 1, 2, 3, 4. If he doesn't listen, do it again and end with a tap on the butt. Again, if there's an overreaction just bring him back up to the fence.
Now combine the two by using your hand with the lead rope to pump the air by his head and neck and the other hand uses the whip to encourage the hindquarters. The horse should move both parts somewhat together. The farther along your horse gets, you can just put pressure in the air towards his girth and he should move over. Again, this prepares the horse for the riders leg to move him.
How do you tell him that he just has to change direction by moving his hind legs only (pivot on the fore)
I would typically just work on these last two exercises before the first one because once he has these two solidified, the top one should be easy. Anyways, to get the horse to move his hindquarters and pivot on the front end, this exercise is called yielding the hindquarters. Again, dressage whip of some sort is handy. Lead rope in left hand, crouch down a bit, aim your eyes at his hindquarters. If he doesn't move his hindquarters away just by that, start tapping the air with your whip. 1, 2, 3, 4. Not listening, increase your energy and if still nothing, whack on the hindquarters. He should eventually just go off your body language. And just reverse everything for the other side. :)
And how do you tell him to change direction by pivoting on his hind, as in a reining spin or a roll back?
This one can be a little bit more challenging for horses so I'd start with the yielding hindquarters exercise first. Start with lead rope in left hand with about a foot of rope from the halter to your hand. Whip in right hand. You can start by just pumping the air with your left hand. 1, 2, 3, 4. If nothing, you can bring up your whip and put it in both hands and pump the air by his head and neck. If nothing, you can tap his nose with your hand. Some horses honestly are so disrespectful you have to smack him on the head. So pay attention. If he gets confused and just backs up or goes all over the place, start again. And then reverse for the other side.
Hope this helps!