I think maybe your expectations for friendship -- both for this woman Dee and for your daughter's friendships -- is a bit off kilter.
You're overthinking this. What you see as "she acts friendly in private but not in public" may be just her personality -- maybe she doesn't want to rush over to you in a group; maybe she was brought up to believe that in a larger group, one does not single out others and focus much on them because that might make others feel left out; maybe she is a basically reserved person who is warmer one on one but in groups is more self-contained. You can't know the answer because you don't know her very well yet. You also can't know if on any given day in a group she might just have had bad news, or be having a lousy day, or just not feel like being a chatty friend to you then and there. But if you were to see her alone the next day you might find her telling you what the problem was, and being more open and warmer with you one one one. Again -- you don't know. Why not give her the benefit of the doubt on that?
I know what you mean about being left out of the conversation at the scout tea when Dee and her friend were talking. But is it just possible that they hadn't seen each other in ages, or had something in common that they were discussing? I've been in that position too and I know it feels like an intentional slight, and it may BE a slight - but probably not intentional.
As for the day her daughter came over and "wandered around our house like zombie" and seems to have bored your kids -- you said yourself that the little girl wasn't feeling well; of course she was a zombie, but she seems to have been expected by your kids, and by you, to be more entertaining, or at least livelier. She wasn't well. Her mom should have said no to her playing that day, but it's done; why does it still loom so large in your thoughts? Do you feel that the daughter is also unfriendly? Again -- she wasn't well. The post sounds critical of her for not being made more comfortable by your daughter's efforts and for not being more playful.
Not sure also why you wanted "an acknowledgement that our kids hung out" when you saw Dee another time in a group. What does that mean and why would you have valued it? The kids hung out that day--not you and Dee. Kids hang out all the time; why is there a need to acknowledge it?
One problem I see too is this expectation that kids will just turn up at each others' doors and say they're ready to play. Around here, parents call each other first -- kids don't just turn up; it would create a lot of disappointment if they did, because kids and parents are busy. You say "Dee was a little weirded out that we just turned up" and I would be too -- you don't know if she was in the middle of something she didn't want to stop but felt obliged to stop since you turned up unannounced. It's not unfriendly or less than neighborly to want people to call before showing up at the door; it's hard to say no to a kid who is at the door, and whom your own kid sees standing there. Maybe yours is just a "turn up" neighborhood but it can put people on the spot.
Why does she mean so much to you that you are so invested in securing her friendship or analyzing her motives? We have been to some houses two different times and much more, and are not warm, close buddies with the families in every case. The kids hang out and we talk with the parents a bit. If close friendships form (and in some cases they do) that is a blessing, but we don't expect it, or expect reciprocation of each play date or acknowledgement, in every case.
Your daugher is young and her friendships will shift very frequently. If you worry this much about every mom you encounter through your child's activities and friendships, you're going to burn out and stay upset a lot. Let it go. Get involved in some things YOU enjoy and find women with common interests that way - it's a much stronger bond than just having kids who are in the same activities or the same class at school.