What do Trinitarian's mean ( I'm including myself here) when we talk about the three Divine Persons of the Trinity?
In our temporal experience a person is easy to identify because every person has two eyes one nose and mouth one body and so every person experiences the world through this mechanism. Namely we have the sense of sight, and hearing, and taste, and touch, and through these senses we experience the world.
But personhood is not the senses it's whatever is behind the senses that is perceiving the world. This is a useful insight because we can say that a person is a center of consciousness from which perception can be interpreted. Now some may argue that when someone dies then personhood is lost because the lights go out in the body and there is no longer any perception.
I would suggest that that is a confusion. The interpreter is no longer in the body, but the interpreter who is the person is alive and well even after the body dies. So perhaps the easiest way of understanding what a person is is to understand that a person is a center of consciousness.
This might be a rather simple conception of what a person is, but the important point is that it is an accurate conception so far as it goes. I believe you point out above that God is spirit and that is true. I'm not going to try to delve deeply into how much broader the notion of spirit is than the notion of a center of consciousness but I'll simply note the fact at this point.
We're someone challenged in thinking about a spirit divorced from a body in the same way they were challenged in conceiving of a center of consciousness divorced from a body, because this is really outside of our observable experience so we have to think about this conceptually. Furthermore we are challenged by the idea of a spirit having more than one center of consciousness.
How do we know that we have this part of the concept right? Because when Jesus prayed to the Father he was not suffering from multiple personality syndrome, this unitary God is genuinely more than one person. At this point you might say this raises more questions than it answers, because so far this doesn't seem very unified. This is where I think the other crucial notion of the Trinity comes into play. And that is that God is one in nature, or sometimes we say God is one in essence.
To discuss this attribute of essence I'm going to use the word nature because it's closer to the way that we use language in this day and age. What do we mean when we say that someone is very "good-natured?" That actually can get into a rather long discussion so I'll simplify it. What do we mean if we say that your dog is good-natured? Well, we're talking about a constellation of behavioral attributes that we find pleasing. So if we take that up one level and say the person is good-natured we might say that they have a constellation of behavioral attributes that are virtuous. Basically what we're saying is in any kind of situation ordinary or extraordinary we have good reason to believe in the virtuous response of the actor. Anyone who has observed more than one dog or cat or horse knows that all dogs have a nature that are more similar to other dogs than any cat, and every cat has a nature that is more similar to every other cat than any dog, nevertheless every dog has a nature that is unique to that particular dog, and none other. That is to say no two dogs have exactly the same dog nature. So if we take that up one level, and say that no two humans have exactly the same human nature we immediately recognize this to be true also. Of course this does not diminish the fact that every human nature is more similar, than any human is to the nature of a dog.
So now that we're thinking about what a nature actually is, we can plug this back into the description of the Trinity. We had this problem of how is God unified when we have spent quite a few words discussing the fact that God has three centers of consciousness? The doctrine is God is one in nature or essence. At this point you might be saying, but everything we just said about nature, is that every human has a unique human nature, and every dog has a unique dog nature, and every cat has a unique cat nature, how can the three Persons of the Trinity have one nature? Because were simply stating a fact about God without having to suggest that it has a direct parallel in nature. There are quite a few ways in which God is completely unique. He's transcendent for example. He exists independent of the universe he exists in the universe and outside of the universe before the universe and if there is such a category as after the universe he will exist there too. That is very different from us. If you remove the universe, and don't create some other realm for us to inhabit, you remove the possibility of us too. So in point of fact, we can deal with these concepts quite handily without having to find a parallel in the created universe with which to delineate the concept itself.
God is the singular Spirit with three centers of consciousness, and one undivided nature.
Since the gospels are the good news about Jesus Christ, we should say something about this third Person of the Trinity. For clarity, when I'm speaking of the third person of the trinity I'm going to say "The Eternal Word." And when I'm talking about that human nature I'm going to talk about "Jesus of Nazareth." So Jesus Christ is both "The Eternal Word" and "Jesus of Nazareth." So in Jesus Christ we have both the human nature of Jesus of Nazareth, and the undivided nature of God in The Eternal Word.
I think this is a pretty good start. With all of this in mind we could address any element of your OP that you think is most prescient.