Chinchilla's ears are very important to their thermoregulation. Since chinchilla's normal environment are the mountains of South America, they are resistant to cold and very susceptible to heat related issues, such as heat stroke. The chinchilla's ears function to expel excess body heat to keep the chinchilla cool. Since your chinchilla now has only one ear, it may be more difficult for her to regulate her body temperature, and therefore you must be more cognizant of the proper ambient temperature if you want her to stay healthy. If you don't use air conditioning please use a fan near her cage. Chinchilla's with 2 ears have great difficult tolerating air temperatures above 75 degrees F, so you must be extra careful with the room temperature. I would keep an extra thermometer near her cage. As far as infection goes, this is always a possibility. Keep the wound area very clean. Place triple antibacterial ointment on the wound. However, if it begins to ooze, look infected, or if your chinchilla begins to be lethargic or inappetant you must seek veterinary attention for the infection.
Lastly, dog's and cats may think of chinchilla's as prey animals, so, as you have discovered, if may not be safe to mix them, even with supervision.
Never let this forum substitute for your own common sense, if you think something might be urgent.
Sometimes Dr Cheng is away at workshops or meetings, and can't get back her for several days.
If your animal has been injured, and you are wondering if it needs veterinary care, the wisest course is to have the animal checked.
I understand about finances (if that's whats stopping you), but Jesse'sgirl is right. A vet would want to see the animal, assess the damage, and probably prescribe antibiotics.
I am of course not the vet, but yes I would say it's a bad thing that your chinchilla's ear got bitten off. You should probably take it to a vet immediately ... make sure it does not get infected and have it examined to see how extensive the damage is. I hope everything turns out to be ok for both of you. :)