There is no requirement that you file jointly any more than there is for United States citizens. I have assisted clients with this type of issue and provide USCIS with documentation of the percentage of married couples who do not commingle their finances. It has not been a problem in the past and if the marriage otherwise satisfies USCIS that it is genuine, you should not have one either.
It's not easy to answer this question, as you don't provide quite enough information. I highly recommend you have a qualified immigration attorney review your case quickly, as the appeal time is very short.
Your question is not completely clear. If your wife petitioned you for permanent residency she is required to complete an I-864 regardless of income, or lack of it. If her income is not sufficient to meet the requirements listed on Form I-184P, then you will need a separate cosponsor, who will complete their own I-864. Hope this answers your question. If not, or you are still not sure, I recommend you contact an immigration lawyer.
Even though your husband has no income, as the petitioner he is required to file an I-864. If he is unemployed say so, and if his income for the last three years was $0, that's what you put. Seems silly, but there you go. If you have more questions, I recommend you get the help of an immigration attorney to respond to the USCIS letter, as you only get one chance to fix the issue they have raised.
Without knowing why you are removal proceedings, it is hard to answer your question here, and you probably shouldn't post details in a public forum like this. I join my colleagues in recommending you contact a local immigration attorney experienced in removal work. I wish you every success in your case.
Evidence of a joint mortgage on your home is just one example of the many different things you can use to show that their was a genuine marriage at the beginning. I strongly recommend you contact a qualified immigration attorney to assist you.
I agree with the earlier responses, but she should not be using your photo without your permission. If she does not respond, I think it appropriate to contact the State Bar for assistance. I would never post a clients photograph without permission.
If your wife obtained citizenship based upon 3 years of permanent residency rather than the 5 years normally required, then this may be an issue. The law requires that she be married to the same person she is using to qualify under the 3 year rule at the time she becomes a citizen. She would have to disclose the divorce at the swearing in ceremony. She should contact an immigration attorney.