Private pay for AL and the lookback period. Will I run into any issues?

This is a very slippery slope, I highly suggest seeing a CELA (certified elder law attorney) before doing anything. It would b better to have things set up and not have surprises at any point of the game.
You need to make sure mom is cared, but also need to make sure that you are doing what is right with the money that is hers and yours. I do know, if you are on joint accounts your money is her money and visa versa. TOD is good, but better safe than sorry. Please seek professional advice.
"it's easier to stop something from happening in the first place than to repair the damage after it has happened".

Yes, opening a separate account is what you need to do. The house needs to sell at Market Value if Mom may need Medicaid within the next 5 years. So I suggest having an assessor evaluate it.

The reason for a separate account is because Medicaid asks for 5 years of bank statements. So keep her present acct ones too. By having a separate acct all that shows up on the statement is what has been spent on Mom alone. My Moms bank, for a small fee, included copies of her checks so the caseworker had them to refer to if he had questions what a check was written for. Do not "gift" any large amounts of money. When I paid out if pocket for something for Mom, I waited till the eom and reimbursed myself taking all the receipts, putting them in an envelope with a check number on the envelope. Yes, keep good records.

Mingling money is never a good idea. Also, any accounts Moms name is on is considered her money by Medicaid unless proven otherwise. My Mom was on my nephews acct and her lawyer suggested removing it because of Medicaid.

When I moved my mom in with me almost 3 years ago I sought the advice of an elder attorney. I opened up a checking account in both of our names. This was the account I had her Social Security go into . also any money that she had from her old bank was added to this account. She had several old small life insurance policies that I liquidated and the money went into that account. So I had about $25000. He told me to open up another checking account in my name and put about 20000. In it. The reason was in the event my mom would have to apply for Medicaid...after 5 yrs ..which is the look back period..Medicaid would not be able to take that money. BUT he said if Medicaid had to be applied for within the 5 years starting from the date I opened that account in my name..that I would need to put that money back into the account with both our names on it. Otherwise Medicare would consider that a gift from mom to me and She would be penalized so many months before she would be accepted for Medicaid. Also in the meantime...if she started running out of money in her main account I could take money from this 2nd account and transfer to her main account.
Then under his advice I opened up a 3rd account only in my name. He said I could charge her rent and this would also be money that Medicaid could not take. I had to have a realtor come in the house and assess her living space. He composed a letter stating what the fair market value was of her living space. He said it was $750 a month. So every month I had $750 transferred from her main account to this account. I stopped the transfer when I placed her in AL.
Medicaid told me when I get down to $5000 that She could apply for Medicaid but her hospice social worker told me yesterday to apply now to see if she can get preapproved. She has about $21000 and after that is gone she will have no assets.
You learn a lot on this journey.

Do not run the proceeds through your account. Setup a separate account in mom's name for that purpose.

My feeling is ALWAYS that the safest way to manage finances is a checking account with LO’s name and POA(s) as well.

In my present caregiving situation, I’m also her designated recipient for her social security.

The checking account, accessible to both POA’s, is used for NOTHING except LO’s expenses, and is therefore an immaculate and unchallengeable recording of her expenses and the payments made for them on her behalf.

I am sure you will get much better answers but..
I would first talk to an Elder Care Attorney.
My first gut instinct would be to open a different account so that the money from the sale of the house is not co-mingled with your assets.
Side comment though. With the diagnosis of dementia would Memory Care be a better option for her?
I would also talk to the Social Worker or Director where you plan on having her and ask if they accept Medicaid and how long would the person have to be private pay before Medicaid.

I’d suggest also that you open up a new account that gets all the proceeds from the house sale $. I would have it as a new checking account with both names on it and on the actual checks.

Not into an old account that is in your name and tied only to your SS#.

Having the $ go into your old account on the initial glance looks like gifting and really IMO you to avoid that red flag ever waving for Medicaid. The account should be POD / pay on death to you so should she die, the $ passes to you outside of probate. Again, should something happen to her and she passes unexpectedly, the $ if POd goes to you. It provides for the situation mom wanted to have happen by having your name added to the title.

Should there be costs to put it on the market, those are costs you need to think about as how to pay. Depending on when mom added you on and how long she is able to private pay for AL and if Medicaid does a 3 yr look back if they are private paying for a facility, you may find yourself in the unique position that there could be no gifting transfer penalty and there is still your 50% left in the account. If you have the wallet, I’d suggest that you each pay 50% of whatever expenses the house has from the day she moves out until it is sold. It establishes that you are a co-Owner and share property costs equally. So if plumber is $1890.50, each of you write a checks from your each individual bank account for $945.25. It helps establish that it is co-owned. Hopefully it’s not on the market long, so it keeps all costs to a minimum.

if the current tax assessor value seems to be whack, I’d suggest that you get it first inspected and then the inspection report goes to an licensed and registered appraiser. They will give you a report with their signature and seal. It is considered legal. So if the house appraisal comes in at below what the tax assessor has, the appraisal value overrides tax assessor. Even if it is significant. It gives you a legit way to sell at a lower price with no repercussions from Medicaid. If house is like lots of elders homes, it has decades of delayed maintenance, is not a new fresh renovation with granite countertops and rainfall shower heads like the comparables probably are. The tax assessor places value on recent sales, so I’d it all crisp new renovations, her homes value will be wrong.

Good luck in all this. Keep a ledger going and keep all receipts as well.

Wait a minute. If the house is also in your name, only half of its value counts as an asset of your mother's. The AL cannot collect more than that.
If the deed was done five years ago or more (the Medicaid lookback period is five years), the same applies. You will not owe all the proceeds from the sale to either.
Talking to an elder lawyer is a good idea. Checking out AL facilities that also have upgraded care (an area that is a nursing home/memory care) might be a good idea too. This way you won't have to move your mom to a different facility if her care needs increase.

Propbably not if your records are perfectly kept but if I were you, I would open a Special Bank Account for this purpose only and not use your regular Bank Account.

contact a good Elder Care Attorney asap and they will help to protect some of the assets and also will help with medicaid when the times comes. make sure you tell the attorney everything so they know what they are dealing with. They will ask for certain things going back 5 years (I know, i just provided all this for our mother), that means bank statements (and ALL pages even if blank), etc. I wish you luck.