I can't answer a couple of the questions that you have here (I don't know if anyone can). I hope I can hazard a good enough answer on what remains, by providing some context for the sources we have. What I will try to do here is talk about what we know about Pompey's visit to the temple and how we know it, what we know about the immediate aftermath of his visit, and then I'll touch on what Pompey "would have expected" to see in a temple.

The two main sources I know for this event are Josephus and Tacitus, with a couple questions potentially sourced to the Book of Ezra.

From Tacitus's Histories Book V Chapter 9 we read:

"The first Roman to subdue the Jews and set foot in their temple by right of conquest was Gnaeus Pompey;​ thereafter it was a matter of common knowledge that there were no representations of the gods within, but that the place was empty and the secret shrine contained nothing. The walls of Jerusalem were razed, but the temple remained standing. Later, in the time of our civil war...[Goes on to talk about what happens 30 years afterward]"

Now, the focus of the Histories was 69-90 CE. So here he is laying the background for Titus Caesar Vespasianus and his siege of Jerusalem, the destruction of the temple, and so on. The section in Book V about Titus around Jerusalem breaks off with the beginning of the siege, then the rest of the book talks about events in Germany. All books after V are lost.

From Josephus's Antiquities of the Jews, Book XIV Chapter 4 we read:

"and no small enormities were committed about the temple itself, which, in former ages, had been inaccessible, and seen by none; for Pompey went into it, and not a few of those that were with him also, and saw all that which it was unlawful for any other men to see but only for the high priests. There were in that temple the golden table, the holy candlestick, and the pouring vessels, and a great quantity of spices; and besides these there were among the treasures two thousand talents of sacred money: yet did Pompey touch nothing of all this, on account of his regard to religion; and in this point also he acted in a manner that was worthy of his virtue. The next day he gave order to those that had the charge of the temple to cleanse it, and to bring what offerings the law required to God; and restored the high priesthood to Hyrcanus, both because he had been useful to him in other respects, and because he hindered the Jews in the country from giving Aristobulus any assistance in his war against him."

Josephus's focus writing here was the event itself, not something a hundred years later, so it is not surprising to find more detail here. What is surprising is that Josephus goes out of his way to say nice things about Pompey, to mention that the temple did have great wealth within it, and that this wealth was not taken.

Shortly afterward, in Book XIV Chapter VII, we read:

"Now Crassus, as he was going upon his expedition against the Parthians, came into Judea, and carried off the money that was in the temple, which Pompey had left, being two thousand talents, and was disposed to spoil it of all the gold belonging to it, which was eight thousand talents.... It was the priest who was guardian of the sacred treasures, and whose name was Eleazar... when he saw that Crassus was busy in gathering money, and was in fear for the entire ornaments of the temple, he gave him this beam of gold as a ransom for the whole, but this not till he had given his oath that he would remove nothing else out of the temple... yet did Crassus take away this beam, upon the condition of touching nothing else that belonged to the temple, and then brake his oath, and carried away all the gold that was in the temple."

Here, Josephus establishes several things. First, the temple had a great deal of wealth in it, being everything that was there when Pompey left it. Second, Crassus took everything that wasn't nailed down and a beam of gold that was.

I think it unlikely that Josephus thought Pompey raided the temple and blamed someone else. In works of antiquity, often a poet or writer will appeal to common knowledge, or say "everyone knows X". To take Tacitus's statement about Pompey in the temple for its rhetorical effect, we are led to think that, at least, highly educated people on the subject would know "Pompey went into the temple, didn't think much of it, and didn't take anything."

In Plutarch, neither the Life of Crassus or the Life of Pompey mentions Jerusalem or the temple. All Plutarch says about Judaea in either Life is in Pompey "he also subdued Judaea, and made a prisoner of Aristobulus the king" and then it appears as in a list of conquered nations.

In Cassius Dio, Roman History XXVII Chapter 17, as he is describing the Jewish people generally we read:

"They are distinguished from the rest of mankind in practically every detail of life, and especially by the fact that they do not honour any of the usual gods, but show extreme reverence for one particular divinity. They never had any statue of him even in Jerusalem itself, but believing him to be unnamable and invisible, they worship him in the most extravagant fashion on earth. They built to him a temple that was extremely large and beautiful, except in so far as it was open and roofless..."

So, what would Pompey have expected to find in the temple? So, to a Roman the word "temple" meant more of the consecrated district, but I'll continue using it in the common sense, that of "the building where all the religious stuff is". The word for "place a god lives" is "aedes". The thing that every Roman would expect to find in an aedes was an interior chamber, called a cella. Inside this interior chamber would be a statue or image of the god. For an example of this, we have the (miraculously preserved) statue of Winged Victory from a temple in Brescia. It's about 7 feet tall.

So Pompey goes in, walks around. He sees gold, he sees spices and tapestries and priests, he sees a cella. He knows that he would never be permitted to look upon the god in normal circumstances, but he is a conqueror and that gives him every right. He goes within the cella and then... there's no god. No statue, no image, no carving, no nothing. It's just an empty room. This is incredibly weird. Everything else about the temple is normal. He says to himself "Okay, I'm in the aedes now, now here's the cella, and now... there's no god!?." It's called a temple, it has priests, it has altars, but it is empty. No one lives there.

To address the fate of the Ark of the Covenant, I hope it is good enough to say "no one knows". In Ezra 1 we are given a relatively detailed breakdown of the treasures taken from the temple by Babylon and then returned to the Jewish people by edict of Cyrus. This treasure gets mentioned a couple more times, eventually being weighed to confirm that everything sent from Babylon had arrived. There is no position in historical analysis or in religious tradition that holds that this treasure contained the Ark. There are differing theological explanations for the current position of the Ark, but none of them involve being given back by Babylon- there are theological implications for the Ark not existing, so these explanations do not include "it got melted down".

If such a vital part of the Jewish religion was missing, why did they not record this when they built the 2nd Temple after they returned from Babylonian captivity ~70years later?

I do not mean to be difficult, but questions like "why did they not record this" can never be given satisfactory answers. To use an example relevant to sources used here, we have lost much of the work of Tacitus. We have lost almost everything that was ever written in antiquity. All we can say is that we do not have any record, historical or religious, about the literal position of the Ark on this planet after the Babylonian captivity. I will note that if the Ark had been returned to the hands of the Jewish people by Babylon, it would be absolutely unthinkable for this event to not be mentioned in the chronicle of the building of the second temple. They listed out the number of bowls, they would presumably have given a sentence or two to the Ark of the Covenant.