First of all, I do not know much about the Medicaid rules. The following link will take you to a website that has some good info and you may be able to find your answers for your Medicaid questions here. These rules are tougher to understand because each State can also have their own rules in addition.
http://www.cms.hhs.gov/MedicaidGenInfo/I have almost 30 years experience in working VA claims and I will share some of my thoughts regarding your VA questions. I think it is important that people are given straight answers when making such tough and important decisions.
Since I do not know all of the facts surrounding your particular case, I must answer in general terms. It shouldn't matter but it does depend on the State that you made your application from. I've seen some VA Regional Offices process and complete an application in as little as 6 weeks. There are some that take over a year. If the application was completed properly and all required documents presented with the application will help speed the processing time.
If the medical documentation indicated your Mother may have problems handling her own affairs due to a mental condition then the VA may send out a 'field rep' to interview her and if they recommend a fidicuary for her, they would have to interview that individual also. The VA does not accept court appointed POAs.
A congressional office can be of assistance but it depends on that particular congressional office as to how much assistance that they 'want' to assist.
If your Mother passes away before the VA makes a determination on her claim then the estate will not get the amount due her. This too is unfair.
I think you have to makes some tough decisions regarding her life expectancy and the cooperation you are receiving from the VA, Medicaid and Congressional Office. If you have any particular questions that I may be able to answer, please let me know or contact the web administrator for my personal email address.
Wishing you the best with your Mother's claim.