I have a few questions regarding encouraging some independence for daily activities and when to provide full support?

You are not just ‘able to make suggestions’. You can set the terms on which your 72 year old mother can live with you. It’s important to work out your terms and your boundaries ASAP, because it’s hard to change things once they are in place. I’ll leave it to others to start on the long list of things to do, things to avoid, as I’m really tired and am about to go to bed early in this very different time zone.

Yes, I would start with her PCP. Get a good physical, check up on chronic conditions, screening tests for mood (depression/anxiety) + a short cognitive baseline. Obtain referrals as required.

Other ideas may be Physio to improve mobility. Some clinics also have OT who can advise on improving methods for ADLs + advise on equipment if required. (Both PT & OT are very good at working out when people won't do a task, as opposed to can't).

Also, I would strongly advocate for anything that she can do for herself - please let her do it. I think a lot of us have this urge to help as much as possible because we think it is helpful and honestly - it's not. It LOOKS helpful but what it really is - in the long term - is detrimental. Mentally and physically. Whatever she CAN do she should do. The phrase "use it or lose it" is very important here. As long as she can do something for herself - she should be - even if it takes her forever and even if it is frustrating for her and even for you. Because that thing -whatever it is - may seem insignificant - but it's actually very important.

One thing that stood out for me was the issue with your daughter. Not only is mother making her uncomfortable, it sounds like mother sees (and maybe treats) your kid as disrespectful and hateful. Pretty soon your daughter will hit puberty and the last thing she needs is someone critical of her in the house. Nor should she have to be uncomfortable in her own home! Home should be your kid’s safest and calmest place, and now it isn’t.

If mother is clinically depressed, she is not going to ‘wake up’ and realize your daughter is a good kid.

Letting it slide because she’s old, depressed, whatever isn’t valid. All your kid knows is she’s being picked on in her own home, and you’re telling her to suck it up. If you don’t defend her, who will?

Making a choice is hard, but you must always choose your daughter first!

As I see your question there are several issues that you present.
1. Your daughter is uncomfortable around your mom. Why is that? Did she have contact with her prior to having grandma move in? Did grandma take her room? Are you paying less attention to your daughter now?
2. Who are you trying to get counseling for? Your daughter? Your mom?
With your mom's diagnosis of dementia counseling is not going to do much good. She is not going to comprehend or retain any of the therapies presented.
For your daughter she may want to talk to someone about a stranger moving into her house, one that is taking your attention away from her, and one that is going to decline and need more and more help.
3. Your daughter is an 8 year old. I have an 8 year old grandchild and it can be exhausting spending a lot of time....if I had to live with with them not sure I could take it for long. And as you know 8 year old's quickly turn to pre-teen and that is a whole 'nuther animal....

I hope you plan on getting help. Your mom will need more help sooner than later. You will be back to changing "diapers", cleaning soiled bedding, and having to be with her 24/7 since she will not be able to be left alone. And PLEASE do not expect your kids to help out. Not fair to them. If they WANT to help great but please do not expect it.

OH, getting to one of the things you asked in your post..
Encouraging independence for daily activities.
Let mom do as much for herself as she can.
When she begins to get confused about what to wear limit the items in her closet that she can chose from. 1 or 2 tops, 1 or 2 pair of pants.
As incontinence starts (unless it has) replace all her underwear with disposable pull up type. When this begins start monitoring her in the bathroom.
When she goes in to the bathroom for her shower or bath monitor that as well, many people with dementia begin to not want to shower or bathe because it can be frightening, confusing for a variety of reasons. Make sure you put in enough grab bars. If there are towel bars near the shower or bath remove them so they are not used as grab bars or replace the towel bars with grab bars.
Remove throw rugs (trip hazard)
Make sure stairs are clear, make sure the floor is clear of anything she may trip on.
this is just a start.

Oh, you might want to look into Adult Day Care for mom. It will give you a break and get her some stimulation and socialization.