Can we get my mom's social security transferred to her while she is living with us?

I would call Social Security. You will need a separate bank account for her SS to be deposited to. Do not mingle her money with yours. Once you give SS the new banks routingc# her checks will be deposted there. If Mom can talk to her other bank, have her give you permission to talk to an officer. I would ask for a printout of any accts she has with them. This would include CDs and IRAs. Make sure you get acct#s. This way if this ends up permanent, you can have things transferred to new bank. CDs may have to wait till rollover. IRAs are different because they are tax deferred.

Your Mom can easily transfer her Social Security to your State. Assist her, or have a bank assist her with making an online Social Security site.
If she is mentally capable she can also make you her POA. Before you do this UNDERSTAND how to act in this capacity. Just today we are seeing posters who are POA who do not understand the importance of keeping receipts, records, all monies in the senior's OWN NAME, how to sign and etc. This is an important fiduciary function with legal ramifications.
You can then arrange with your Mom how you wish to be paid for room and board; be certain this is a care contract overseen by a lawyer so that it has IRS consequences for you and for your Mom, and so that if in future she needs medicaid her chances are not ruined by having considered to have GIFTED money to you.
This is going to be a WHOLE LOT on your plate. You are going to have a huge learning curve; my first year acting for my brother was an anxiety provoking nightmare and I wasn't even in charge of his care bodily; he was in an ALF.
I wish you a lot of luck. The fact that your brother won't give you any financial information is quite worrisome. You would do well now to see a lawyer.