The Assyrian king Sennacherib gives an account of the siege of Jerusalem during the time of Hezekiah, which is also attested in the Hebrew scriptures. Naturally, each have their own point of view, with both claiming a great victory. The biblical account does invoke supernatural forces, with an angel killing 185,000 Assyrians in one night. Not surprisingly, the Assyrian account has no mention of this.

That would be around 700 BC.

The Mesha Stele provides claims on Israelite king Omri c.850 BCE. It provides an account of some of the events of 2 Kings 3:4-8.

See also this comprehensive list of artifacts significant to Biblical stories.

I am unaware of any independent verification of the stories prior to David, c.950 BCE. The Patriarchs, Egypt, the Exodus, the conquest of Canaan, the period of the Judges...pretty much everything from Genesis 11 to the end of Judges is sort of up for grabs.

The flood, definitely. There's a 3500 year old Mesopotamian tablet containing the "Atrahasis" account of the flood. Details and names were different, of course. And there wasn't one god, it was various gods.

It's one of the most interesting pagan tales out there (in my opinion). Because after the flood, the gods decreed that to avoid other floods they would limit human lifespan, and bring diseases and miscarriages to regulate the population.

This has a very interesting parallel with the biblical account of the flood, because right before the flood, there's a passage where we read that God's spirit would no longer dwell in men, shortening their age to at most 150 years, IIRC 120 years.

Perhaps you might be interested in other parallels. For example, the confusion of tongues has a parallel with the Babylonian epic "Enmerkar and the Lord of Aratta"(http://en.wikipedia.org/wiki/Enmerkar_and_the_Lord_of_Aratta).

There are more vague parallels between the Bible and Ugaritic (Bronze Age Canaanite) texts. Please check out "The Ugaritic Texts and the Bible" by Jerry Neal. Some of the Ugaritic Texts give insight into obscure passages of the Bible. For example, the Song of Deborah in Judges 4-5 mentions a Shamgar son of Anat. Anat was a Canaanite goddess, sister of Baal. Several parallels between goddess Anat and the heroine Deborah are explained in that book.

In Exodus 32, the grinding of the golden calf bears a resemblance to Anat's defeat of the death god, Mot. (who killed her brother Baal; and by killing Mot, death, Baal was resurrected). There are more parallels, but the resemblance between biblical stories and Ugaritic tales aren't obvious, and sometimes learning Hebrew and Ugaritic is required to see the resemblance. But that doesn't make them any less interesting.

EDIT: More details.

There was a recent feature on the Biblical Archaeological Review website: a supplement to the article "50 People in the Bible Confirmed Archaeologically." This isn't exactly "stories" in quite the sense you're looking for - especially not in the sense of records of literary-mythological people/events like the primeval ancestor Abraham and David's battle with the giant - but it would be a nice place to start, if we're really looking for the earliest things here.

Around the 9th century BCE is basically the starting point for all this. E.g. you can find reference to Shoshenq I's Palestinian campaign (later 10th century BCE) in 1 Kgs 14:25-28; 2 Chr 12:1-12. And someone has already mentioned the Mesha Stele (which I commented on at length here).

For David and Solomon, I'll defer to other people/threads. If you look through /u/Flubb's post history, I think you'll find a lot of stuff on this. /u/ScipioAsina has made some posts on this too, e.g. here (I'm highly skeptical of the historical value of some of this, though they're involved in original research on the topic, so I'll reserve judgment until that.)

Some scholars have seen, in the Exodus story, garbled or faded hints of the West Semitic Hyksos occupation of (and subsequent expulsion from) Egypt in the 17th century BCE - though this is far from scholarly consensus. However, we know that the memory of this survived for millenia; and we find a conflated version of this + the Biblical Exodus story in the early 3rd century Greco-Egyptian writer Manetho's Aegyptiaca...but this doesn't necessarily speak to anything about the original Biblical account.

Great question! This ‭is one that first got me interested in Biblical scholarship. While this is generally covered in others' links, let me try to give a quick-and-dirty summary of what I've gathered over the last couple decades of dilettantish study:

The earliest Biblical character that can plausibly be called "historical" is king David, based on the Tel Dan stele's apparent reference to a "house of David" (Solomon's got no support; and I'd say he's basically a personification of the city of Jerusalem). King Omri of Israel (Shechem) is the first to have serious, indisputable extrabiblical support, though he's given short shrift in the Bible itself. This doesn't count Hiram of Tyre - who was quite well-known, but just thrown into the Biblical narrative to add "credibility".

Before that, you can draw parallels between any number of Biblical and extra-biblical stories, if you're just looking for literary influences: perhaps most notably, the first creation story of Genesis seems to be based on the Babylonian Enuma Elish; and the flood story has many earlier versions. Exodus may be loosely based on the expulsion of the Hyksos from lower Egypt; and the "Hebrews" in general are probably based on the Bronze Age "Habiru" (which referred to Levantine nomadic peoples of any religion or ethnicity), though you'll find a lot of apologetic arguments to the contrary.

You may be interested by this very good post on David and Solomon. And just to add something constructive: Biblical accounts of the patriarchal age (Abraham & cie) are very debated—some scholars think it preserves a kernel of historical truth, while others think that they were just later mythical creations, that may have reflected the elite rural world of the time of redaction, but had nothing to do with Bronze Age Canaan (Wellhausen's hypothesis).