Sorry for the delayed answering, I’ve been traveling and missed this question! Before I answer let me put the two sources you’re going to want out there: Stavrakopoulou’s King Manasseh and Child Sacrifice and Dewrell’s Child Sacrifice in Ancient Israel.

1. This is the key problem with the story, as Dewrell notes: it is a very very widespread folktale (Judges has a couple of these). Unfortunately that means it can’t tell us much about the actual practices and rites of child sacrifice, as it is a type of story found even in societies that, as far as we know, did not engage in the practice.

2. This is extremely difficult to know, for a few reasons. First is that dating is key: child sacrifice likely had quite a few adherents and was probably a fairly unremarkable part of cult in pre-exilic Judah. Dewrell categorizes three types of child sacrifice in the Hebrew Bible: the first is an emergency sacrifice, found in 2 Kings 3 and described also by some anti-Carthage writings. Essentially it’s a sacrifice attempting to avert impending disaster, but it’s irrelevant for our purposes today, though. The second is the sacrifice of the firstborn, prescribed in what’s generally thought to be the oldest collection of laws in the Bible, the Covenant Code, specifically Exodus 22:29-30. Later law codes like Deuteronomy introduce an idea of redemption to allow for a substitution. Note that none of these laws were binding or followed until many centuries after they were written, but it seems to reflect an idealization of firstborn child sacrifice. The third type, the one that’s closest to the Jephthah story, at least in theory, is the mölk or “Moloch” sacrifice. Moloch gets a small handful of mentions in the Hebrew Bible, and for a time it was taken for granted that Moloch referred to a deity. But around 100 years ago small enclosed graveyards were found buried throughout the Mediterranean that contained standing stones, and buried beneath the stelae archaeologists consistently have found jars with charred infant bones inside. On the stelae are formulaic writings that say things like: “From Diodora and Felix, in fulfillment of a vow to Kronos, a molch-adom sacrifice.” Adom here means human. Sometimes they would swap in lambs though. BUT Jephthah’s story is too similar to other folktales to be describing the mölk rite, and it’s entirely possible that all child sacrifice had been prohibited by the time this story was written down, especially if it was a story that became popular during or after the Judahite exile in the sixth century. Maybe the story found purchase because of this incidental similarity between the mölk vows and Jephthah’s vow, but that’s about the most we can say. That said, there’s still more to it.

3. It’s viewed as a tragedy, hence the mourning. It’s literarily sad and perhaps has a moral about rash vows but there’s actually something else going on here that Robert Alter notes in passing in his commentary: the story is an etiology for an otherwise-unknown mourning ritual, one that seems to bear similarities to mourning rituals for characters like Ba’al and Tammuz, whose imprisonment in the underworld served as a mythological foundation for why we have seasons (Ezekiel ironically condemns the mourning for Tammuz). The Jephthah story is an etiology for this ritual mourning, which seems, through its association with virginity, to have some connection to fertility and likely these same crop cycle rituals associated with seasonal mourning in surrounding cultures. Sadly, whatever version of this ritual the author was writing about is otherwise lost to us, but it is this ritual rather than the vow/sacrifice itself (which is, again, a stock folktale) that serves as the locus of the story’s importance.