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Existence
Philosophers

I have been reading Bertrand Russell's History of Western Philosophy , and am puzzled by a paragraph in a section on Plato ('Knowledge and Perception in Plato') pertaining to the use of the verb 'to exist'. The paragraph reads as follows: "Suppose you say to a child 'lions exist, but unicorns don't'; you can prove your point...by taking him to the zoo and saying 'look, that's a lion'. You will not...add 'and you can see that that exists'...if you do then you are uttering nonsense. To say 'lions exist' means 'there are lions', i.e. 'x is a lion' is true for a suitable x'. But we cannot say of the suitable x that it 'exists'; we can only apply this verb to a description, complete or incomplete. 'Lion' is an incomplete description, because it applies to many objects: 'the largest lion in the zoo' is complete, because it applies to only one object". What puzzles me about this paragraph is quite how it is, as Russell sees it, nonsensical to say 'there is a lion, and it exists'. Is it because we do not...

Russell may be making the claim that existence is not a property. An individual may have the property of being furry, of making loud sounds, and of living in Regent's Park Zoo, but it does not also have a property of existence. Rather to exist is for those properties to be instantiated. To say that something exists is to say that there is something with various properties, but existing is not one of them.
Existence

What is the reasoning behind the existentialist claim that existence precedes essence?

I'm no expert in this area, and I don't know what the reasoning behind the claim is, but as I understand it the meaning of the slogan is that people are not born with an essential nature, but must choose their own identity. (Did Sartre have any children?)
Existence

Even at the lowest levels of proof does not the existence of something in one's imagination give it at the very least a semblance of actuality?

I'm with Alex: I can imagine a mountain made of pure gold without that mountain existing, even a little bit. But it may well be that my act of imagination entails that there must exist something else, namely that cause of that act. Surprsingly perhaps, Descartes used this line of thought for one of his arguments for the existence of God. He had an idea of God, and he took that the idea must have a cause, and that the only possible cause in this case is God Himself. Why? Because a cause must have at least as much 'reality' as its effect, and only God has as much reality as the idea of God. Not, it must be said, a very convincing argument to modern eyes: why can't 'big' ideas have small causes?
Existence

Is it possible that such a thing as a "ghost" exists? Not the scary scooby doo type ghosts, but some sort of lingering remains that could haunt or protect over. My friend's aunt claimed her whole life to be able to see angels and demons, with the angels the only thing holding back the demons from getting to us humans. She is one of many that I have heard about with this sort of thing. It's in the same ballpark as the ghost question, can these things possibly exist?

Disembodied minds may be logically possible, but I think there is no good reason to believe they are actual. It's far more likely (though more boring) that your friend's aunt was imagining those things than that they really exist.
Existence
Knowledge

I believe that I am the only thing that really exists. I think that my friends and people I meet are versions of myself if I had taken a different path in life. I could be anyone and I can understand even the most ridiculous of ideas. It seems like a negative view but I am convinced that everyone or everything I encounter is to benefit me in some way. I don't believe in good or bad. Nor emotions or science. Just nature. I was created and all I am here to do is survive as long as possible. Period. No silly questions about the meaning of life or what is my purpose or am I a good person. Life isn't a gift it was just something that was possible and eventually happened. I think people like to lie to themselves to forget the fact that they are basically useless. I apologize for making this sound negative and too long. I guess my question is how can anyone prove to me that they really exist?

Your question reminds me a little of the story Bertrand Russell told about the philosopher who claimed that solipsism -- the view that only you exist or anyway that there is no reason to believe anyone else exists -- was obviously the correct position and she couldn't understand why everyone else didn't agree with her! But there does seem to be a sense in which you cannot prove that anyone other than you exists. Actually, in that sense, it also seems that you can't prove that you existed in the past or will exist in the future. So you if you have standards this high, then the most you know is the content of your current experience, though even this may be saying too much.