Actually,
that’s just an urban legend, and a widely-repeated one. We have no contemporary and later sources (not even Josephus or rabbinic literature!) which state something to such effect. The first mention we have of a rope being tied around the high priest’s body or ankle is from the Kabbalistic work
Zohar (13th century AD).
Of course, we need not necessarily think the veil was so thick as the other legend purports, or even that it was so filmy thin. I imagine it being perhaps as thick as a modern-day thick rug, if not slightly more. Not as bulky as is often said, but at least still sturdy to some degree.
We don’t know about that. The Ark, along with other important items, are not listed among the cultic vessels looted by the Babylonians (1 Kings 24-25); so what naturally follows is it could have disappeared at that point or earlier. What became of it? We don’t know.
For the record, the reference to Jeremiah hiding the Ark and other stuff occurs in the context of a private letter (1:10b-2:18), where it is attributed ambiguously to “
the records”. Personally, I have doubts as to its historical veracity - the story appears a few centuries after Jeremiah lived, and it openly addresses a point which so concerned people at the time the letter was penned: that the new Temple is somehow less holy or probably even illegitimate in the absence of the Ark, the Tabernacle, and the Altar.
As for what the High Priest did in the Ark-less Temple during
Yom ha-Kippurim, the simple answer is that he may have performed the usual ceremonies as if the Ark still rested inside. For example, where the blood is sprinkled before the Ark then, he now sprinkled the place where it would have stood.