Why was Jesus not welcomed and carefully tested by the Jews if they were (and are) actually waiting for Messiah? I have tried to understand these things all my life. A lot of the Bible stories make no sense to me, which probably means I'm missing something. One of the biggest questions I have is this:

If the Jews were (and still are) waiting for Messiah, why did they become angry instead of overwhelmingly happy/excited from Jesus' arrival?

Let's assume that there had been numerous false Messiases prior to Jesus coming along. OK. But they are still waiting for Him, right? So why did they not have some kind of well-defined test which must be passed for them to believe that whoever claims to be Messiah truly is Messiah? Why wasn't their reaction: "Oh! Yes! Welcome, potential Messiah! Please just prove it to us right here and now and we will believe you and accept you as Messiah! If you cannot, you will be asked to leave and stop wasting our time!"

And then the person claiming to be Messiah would either fail or succeed, and then they could be either happy or angry after that point.

Why instantly assume that Jesus was a fake Messiah, before allowing him (and others) to prove it (or fail to do so before their eyes, giving them proof that it's a fake Messiah)? I don't understand it. Is the real Messiah said to arrive in some specific manner, and that's the reason why they became so furious when Jesus claimed to be Messiah? What if he had actually been Messiah, and they just crucify him like that?

Maybe this is a naive question, and I'm lacking a lot of information, but I really don't like it when I feel as if crucial information is being kept from me when these stories are retold over and over.
He was tested

The Jews living in the time of Jesus did try to discern whether Jesus was the real Messiah that they had been yearning for, because they disliked living under Roman rule and wanted to go back at least to the Hasmonean period, or better yet, to the David &amp; Solomon period (the golden age).  

They were waiting for God to provide them with what God Himself promised in His covenant with David (2 Sam 7:12-17), confirmed through His prophet Isaiah hundreds of years later ("a shoot from Jesse's stump", Isa 11:1-10). But as the waiting grew to hundreds of years after Isaiah, theories of what form the Messiah would take abounded. One theory even predicted 2 messiahs, one from the line of David, another from the line of Aaron.  False pretenders came and went.  By the time of Jesus the yearning didn't go away, it only became stronger.

Many passages in the Gospel showed how they approached Jesus, observed what He said and did, saw His miracles, heard testimonies including by John the Baptist, asked for miraculous signs, etc.  They were excited for a while, but when Jesus began to claim that He was the Son of God and broke what they thought to be the right way of observing the Sabbath (and many other rules added to the Torah by the Pharisees) they began to have doubts and thought that maybe Jesus was another false pretender.

The lack of a warm welcome

It was because Jesus did not meet their expectations.  

The climax was when Jesus was crucified.  It was a scandal.  A messiah shouldn't die a shameful death.  That meant God was not with him.  The problem was that they didn't read OT the way Christians do.  Many passages in the OT continue to be interpreted differently by the Jews of today (just see Judaism.SE for yourself, such as this answer, to see how different the interpretations can be).  For example, the Suffering Servant prophecy in Isa 53, so critical to Christian theology to identify Jesus as the Messiah sent by God, is interpreted to refer to the nation of Israel which had been displaced and persecuted throughout thousands of years (the 20th century Holocaust only made that interpretation stronger).  The Jews then as well as today don't believe that God resurrected Jesus from the dead.  To them then as well as today, Jesus was either a lunatic or if not a lunatic he committed the most serious blasphemy by claiming to be God.  If Jesus was the latter, by that calculation Jesus deserved to die, since God commanded them to treat false prophets that way (Deut 13:1-5).

I hope reading the Gospels against the background above helps answer all your questions.  Read especially how Jesus responded to them.
Basically, they were too close minded.  They didn't want to believe Jesus's "more than enough" signs and wonders and testimonies, and Jesus &amp; the Gospel writers chalked it up to "hardness of heart", predicted by Isaiah as well (Matt 13:12-17 quoting Isa 6:9-10).