It sounds to me like you need to have a very honest (and non-confrontational) conversation with your mom about her expectations for the dog. If she truly "hates" dogs as you say, that does not bode well. If this is true, I'm surprised that she allows you to keep a dog at all. How long have you had Emma?
I'm also curious about how is Emma currently housed, because some situations are better than others. Is she on a tie-out in the yard? In an enclosed kennel? In your garage or another outbuilding? Does she have a nice dog house where she can get out of the elements?
I think that you need to find out point blank whether or not there is a possibility for the dog to ever live in the house, and then you need to accept that reality once you find out for sure. You say that "you think" your mom would allow Emma inside if you just fix the 2 problems you list, but are you sure? Talk with your mom and get it straightened out now. There is a reason she hates dogs, and there are probably additional reasons to what you list that she doesn't want Emma inside. Be honest and try to get your mom to air all of her grievances to you... for instance:
Does she fear your dog?
Does it maul her for attention/bug her all the time when it's in the house?
Does it bark and disrupt the peace?
Does it make messes inside that she doesn't appreciate - shedding, tracking in dirt, peeing?
Are you taking 100% responsibility for the dog's feeding, watering, exercise, grooming, etc every day... or does it sometimes fall to your mom to care for it?
Does it make your mom mad when she finds out that you've been sneaking the dog inside against her wishes?
I'm not saying any of those is the case, but you need to find out if there are compromises or conditions that can be met where your mom and Emma can co-exist.
If it's the dog's behavioral problems that are the issue, you should explain crate training to your mom as an option, and try to make her understand that keeping the dog outside will never result in better house manners. House manners need to be trained in the house, so the dog learns what is acceptable. And it is much easier to train any new behavior when you start training inside, in a quiet, non-distracting environment. If you crate-train Emma, she will be confined and out of your mom's way when you are not there to care for her.
Why is the bathing issue a problem with your mom? If Emma is dirtying the house when she comes inside, you should try intermediate steps to keep her (and the house) cleaner as you slowly work through her fear of water. Wipe her feet clean when she comes through the door. Use those dog wipes, or try those spray-on, towel-off dog shampoos.
The jumping up can be trained out of Emma with patience and practice (and sticking to ONE method), and you can work on her greeting people calmly. Above all, you need to play and train with her daily to give her the stimulation and human interaction she needs. If you have the commitment to working with her every day and cleaning up after her, have that talk with your mom and see if there is a viable option for Emma living in the house. I'd recommend the crate training, starting with the crate acclimation in her current outdoor living area, and eventually moving it indoors.