I think the neutering will help a little but I think he needs to be corrected on the behavior also. These seem to be small problems just annoying. Maybe you could have a trainer help you. They seem very fixable. In my past experience a leash in the house is helpful until you establish boundries. If he has a leash on and he is excessivly sniffing you can correct him. If you work on basic commands like sit and stay, when he starts to do this pull him away and make him sit and stay in front of the other dog to see this is what you expect. He has to fix these problems or no one will want him
I hope it works out and you can keep him.