Nobody can prove the existence of God because God is immaterial spirit and it is His choice to reveal Himself or not. Not even an Orthodox Christian can prove His existence. All an Orthodox can do is be light for the world, to be so loving and virtuous that those around him will have no choice than believe in God. This is why people believed in Christ also during His time on earth. They were not so moved by the fact that He was indeed visible and among them, as much as by His loving presence and virtue. So, indeed, it was their hearts that were so moved by Jesus, and not their intellects (or that they could prove He was there).
Which brings me to why God requires Faith, not proof. It is because He doesn't want so much that you believe He exists and that you worship Him, as much as He wants that you yourself become like Him, that you change (repentance), that you become loving and virtuous. So, He is after your own benefit, not His own. Otherwise, indeed, He would just "drop from the sky" and say, worship Me, give Me things.
It is a very low thing to ask for evidence. You only want to be satisfied right away, but you don't want to offer and work towards being worthy of God's revelation to you. You feel you are owed everything, but you don't owe anything in return. Worst of all, the fact that you could be loving, virtuous, patient, understanding, etc., without evidence does not really move you; all you want is things (proof). So, make no mistake, it is better to say that you don't know if God exists, but to love a God and to prepare yourself according to the absolute standards of such a God, than to say that there is no proof of God, therefore I will just do whatever. Or, another example: you can say you are not sure about Christ's Divinity, but believe that He really stood for and lived according to the highest level of virtue possible; therefore, you want to be virtuous yourself. Or, instead you say that I don't believe in Christ's Divinity, therefore I think His virtues were garbage? So, it's all about what you want for yourself, what you want to be yourself, not about what you give to God. Do you even believe that God is owed praise and gratitude? Or you really just want things from Him? Do you really love your neighbor, or it really doesn't matter to you since you can't prove God exists? See, God does not force you to do these things, they have to come from you.
Needless to say, proof of Him would ruin our own journey towards freely choosing to be virtuous. How great it is in the eyes of God the virtue of people who did not know He existed, but gave their lives away for their brothers, who raised children, who worked for their country, who did not blaspheme or unnecessarily questioned God's existence! You say there is no proof of God, but it's really how you want to see things. How much better it would be if people admitted the truth -- that they don't know if God exists. How much better it would be if they felt that regardless if they couldn't prove God's existence, they would so love it if He existed, that they could not picture a universe without Him.