How can you prove you exist, if everything you do is a continuation of that lack of proof situation?

"lack of proof situation"? Oh you poor fool, there is no such thing! Your mere attempt is proof! The mere attempt to prove your existence is in itself proof of it. If you seek more than that though, then it becomes proof of recognition, meaning proof that others acknowledge your existence. Existence itself is absolute. It is undeniably true. You exist, and the situational constant is so as well. In other words, you have already proven that you do exist.

https://www.youtube.com/watch?v=wv-34w8kGPM

When I eat a tasty hamburger, it turns to shztt. Therefore, I must be real.

"I think therefore I am"

 what do you mean "exist"?

1) everybody has dreams , YOU know you have had one, but how would YOU prove to somebody else that you did?

If that is important it should be done before you don't exist.  It would be pretty hard to exist after you don't exist anymore.  Probably best to get on with it, make sure it happens while it can happen.  

If such a proof exists beyond the mere fact it requires us to first build a model of that which we experience. And to incorporate what we model of our experience into a complete theory. Even if we lack proof for the theory it is better to have such a theory than not. We need to act with decision. If we spend millions of years waiting and do nothing globally then it will be a waste. As we experience will the grand theory be adapted. It should be imperceptible any flaw. Beyond detectability. We create the meanest and mightiest theory. Like a fortress against all that might bring it down.

It might happen that beyond what science can tell us there will be belief systems. Each one is a venture into what might be. In any case we should always endeavour to evolve our grand theory. Even abandon it if it not as effective. Should we discover a more novel approach. Or a eureka moment. To make a tighter more economic theory.   

Our theory is like a machine we build to serve a function. We measure it’s success by how to fulfils that function. Which might involve multiple objectives in tandem. And we must judge what the most aesthetic compromise is.

These words I type is proof I exist because I’m using my hand on the keyboard. Another way of proving I exist is to meet me in person but not in a dark alleyway since I prefer somewhere safe.

Cogito ergo sum.
It is possible that I am just a bunch of thoughts floating in a vacuum but that is still an existence of sorts.

Ride in an elevator with me this morning. You'll know I exist.

In absence of effective existence by many people, society can easily collapse. They should not be obstructed. Life belongs to God, not man. No requirement to prove existence of its purpose to man.

When dealing with the question of determining self-existence, as you note, either one has Self-realization--I am that I am--Moses' God, and the God of the Psalmist(s), Rabbi Ezekiel, Elijah, and the like--or one properly seeks such realization...as there is One and no second save the mutable existence that you term "lack of proof situation."

Related:
"The Star of Redemption" by Franz Rosenzweig;
"Return to the One:  Plotinus's Guide to God-Realization;"
"There's Something about Godel;"
"The Path of the Higher Self."