There's your answer...carry a ball at all times... when you want something from him, whip out the ball, if he complies, throw the ball! Start simple with stuff he knows, reward him with the ball!
Yep. Food treats are generally easier for obedience IMO (as in, easier for the handler to manage), but if he won't take them, a ball is definitely an acceptable reward.
OP, you talk a lot about corrections...have you spent any time teaching him how you want
him to behave? It sounds to me like he does well in a structured environment with someone who handles dogs professionally so is very good at using his or her body to help cue the dog (training class), but he's out of control anywhere else. What this says to me is that he is willing to listen, but you are not able to motivate him to do so.
I think you are right to not escalate to the e-collar. I personally don't have anything against them when used correctly, but I doubt that if all your other corrections have had no effect, an e-collar is going to work. They can also cause a lot of harm if they aren't used perfectly, and I think most dog owners do not have the experience to do so.
I would probably try another trainer. I would also leash the dog up to me and basically treat him like a puppy or a new rescue and basically re-teach the dog the rules of the house. Focus on teaching him what he's supposed to do, not what he isn't supposed to do. Correct him if necessary, but your goal should be to manage his life so that he doesn't really have a chance to misbehave.
Remember, to train a dog you have to motivate them. Right now, he's not listening to you because there's no reason to--whatever he's ignoring you to do is a lot more rewarding for him than what you can give him. This is where his toy craziness comes in. Use his toy to reward him for listening. I would take away his toys except for when you're training him, actually.
At this stage, reward him every
time he responds to you, or does something you want. I would strongly recommend clicker training for you, since it can be hard to reward instantaneously with a ball. The clicker will mark the exact behavior that you want.
I also agree that right now, you should crate and rotate him and the Dachshund, or at least crate him when your wife comes home and he always goes after the dog. Punishing him like you're doing is clearly ineffective (and depending on why he's going after the dog, it may even be making things worse). You may be able to reintroduce them later, but right now you need to separate him until you reliably get him under control, for the little dog's sake.