Should I be scoping out nursing homes for Mom, as we see cognitive changes (but no diagnosis)? I need her income & expenses first, correct?

Sounds like your hands are tied right now....like your mother still drives and is competent enough to run her own life and wants no interference or "snooping" of any kind. If and when an Event occurs, that's when everything will change. You will be forced to find her a place to live that meets the guidelines set forth by the hospital or rehab facility as a condition of release. Everything will be heaped on you at that time and it's overwhelming. It happened to me in 2014 when my dad fell and broke his hip. The rehab refused to release him back to independent living, even though my mother was there to help. I had been kept in the dark about their finances....it was none of my business, despite being an only child. Dad needed a SNF, but I needed Assisted Living because mom was pretty ok at that time and did not need Skilled Nursing. Truly a cluster.
I don't know what to tell you except to do some research NOW, before the Event happens, and have your ducks lined up. Find out the Medicaid requirements in your state also, should the need arise to apply. In fact, set up a free consultation with an Elder Care Attorney and tell him what you've told us. Let him or her guide you accordingly.
I wish you the best of luck. It really is sad what some of our folks are willing to put us through, isn't it?

First, and I am not sure but sure someone will chime in, but if Mom is not living in her house, the reversed mortgage stops. So she will no longer have that income.

Its a good idea to look into LTC but be aware, she has to need care 24/7. If she is capable of caring for herself with just a little direction, LTC may not be an option. Medicaid may not pay if that is the route u have to take.
There are Assisted livings but most are private pay. This could be 5k or more a month. Memory care is for people pretty much into there Dementia and cost maybe a little more than an AL. Then there is LTC. 10k a month private pay. Medicaid may help when all assets are used for care.

To me, researching and planning is a great idea, be prepared to make the move. My brother and I took months to find the right facility for his father and step-mother. They are now in a wonderful place.

I think that researching facilities, AL, Memory Care and NHs is a great idea.

When the big fall or illness occurs and mom is in the hospital, the discharge planners will tell you what level of care mom needs and if you've scoped out the territory, you will know which ones are acceptable.

As your mom's POA, you should know about her assets and income. This discussion should be entirely separate from the idea of living in a facility. It just makes sense on all levels for the POA to know what she has banked and what she has coming in.

I have 3 kids. My POA kids knows exactly what I have and where the passwords are. The other 2 know nothing.

Probably not a nursing home, instead assisted living/memory care.

Yes you should start looking and find out requirements for her to enter a facility. The doctor has to prescribe it, so not really your decision.

And if mom is considered competent it is still her decision on where she wants to live.

Having her financial information would be helpful, but again, you are not entitled to that information unless mom allow it.

Tread lightly.

You can always call her doc to provide them information on your concerns, but again, they cannot tell you anything unless mom has signed a HIPPA release.

I suggest you meet with an attorney, or look up online about what you can and cannot do as a POA.