The best resource I have found to fully understand this doctrine is from Dr. William Lane Craig’s Defenders podcast. There is audio, video, and a transcript here of the very engaging and enjoyable eight part series: http://www.reasonablefaith.org/defenders-2-podcast/s5
Here are some relevant excerpts that address the OPs main questions:
This doctrine of the Trinity is unfortunately often obscured by Christians in mystery.1 Very often they will simply say that the doctrine of the Trinity is incomprehensible and no one can understand it; it is a mystery and thereby they excuse themselves from having to explain it or think about it. But I think that this is an unfortunate tendency because the doctrine of the Trinity is really not in any way logically incoherent or mysterious. The doctrine of the Trinity is not the doctrine that three Gods are somehow one God. That would be clearly self-contradictory – to say there are three Gods, and these are one God. Neither is it the claim that there are three persons who are somehow one person. That, again, would be self-contradictory – to assert that there are three persons who are all one person. But the doctrine of the Trinity does not assert that there are three Gods that are one God or three persons that are one person, but it asserts that there is one God who is tri-personal. It is one God who is three persons, or, to put it another way, there is one God who has three centers of self-consciousness: the Father, the Son, and the Holy Spirit.
I think it is unfortunate that very often Christians will appeal to inadequate analogies to try to explain the doctrine of the Trinity. There is really no reason to expect that there would be any analogy to the doctrine of the Trinity. There is no reason to think that there has to be some created thing that would so reflect the nature of God that it would be a good analogy. Most of the analogies that are suggested are inadequate. For example, here is one popular analogy for the Trinity: One man can be a father, a son, and a husband. That is like the Father, Son, and the Holy Spirit being one God. I am one man, but I am a father, I am a son, and I am a husband. So I am three-in-one. Well, that is not an adequate analogy for the Trinity because that one man simply has these three different roles, but there is only one person involved there. That is not the doctrine of the Trinity – that there is one person who functions in these three different roles. That is a heresy; that is not what the doctrine asserts. Sometimes it is said that water (H2O) can take the form of either liquid, steam, or ice. It can be either a liquid, a gas, or a solid. That is the way the Trinity is – the Father, Son, and the Holy Spirit are all the same in essence, like H2O, which can be in these three different forms – liquid, steam, or gas. Again, the problem is that the liquid, the steam, and the gas are simply different states of the same substance successively. You can have water being in these different states, but that is not the doctrine of the Trinity. This again is a form of what is called “modalism.” So I personally think that it is better just to eschew any of these analogies. I don’t think we need to have analogies to the Trinity. The doctrine is clear and logically coherent in what it states; namely, that there are three persons who constitute one God.
The way I like to think of it is in terms of self-consciousness. My being supports one center of self-consciousness which I call “I” – that is how I refer to myself. I am one being who is one person. I have this one center of self-consciousness. God is a being with three centers of self-consciousness – three “I”s. There is the Father, there is the Son, and there is the Holy Spirit. So just as I am one being with one center of self-consciousness, God is one being, but with three centers of self-consciousness. I think that is the easiest way to think about what the Trinity is. We will say more about this. This is just by way of introduction. But I want us to get a handle on what the doctrine of the Trinity affirms rather than what it does not.
The doctrine of the Trinity is not something that is explicitly taught in the Bible. The doctrine of the Trinity is a systematic summary of the biblical material concerning the Father, the Son, and the Holy Spirit.2 Theologians take the raw data of Scripture and reflect on it and systematize it. The doctrine of the Trinity is the result of that reflective systematization of the raw material of Scripture. Therefore, it really doesn’t matter whether or not the word “Trinity” is found in the Bible. The name just isn’t germane. The question is whether the doctrine is one that is biblically faithful. Is this a faithful systematic summary of the biblical doctrine (whatever you call it)? If you didn’t call it “Trinity”, maybe you could have called it “Uni-Triad” or something. You could have had some other name. The name isn’t important, and therefore it is of little relevance whether or not the word “Trinity” is found in the Scripture. What is important is, does the doctrine faithfully represent and systematize the teaching of Scripture concerning the Father, the Son, and the Holy Spirit?
Read more: http://www.reasonablefaith.org/defenders-2-podcast/transcript/s5-1#ixzz4ke8Id8fT