Some scholars will tell you that the evidence for a David, even a dynasty founding king David, might be there. Sonny Ordell's answer lists the sparse and weak evidence we have for that.
But the way you phrased the question poses an additional difficulty: the listed qualities attributed to that King David cannot be all true.
The archaeological evidence for the vast territories conquered in the time to be considered are just not there, the exact situation and circumstances for the establishment of the dynasty, its predecessors or the changing status of Jerusalem are all unclear outside of the biblical references themselves.
Sometimes current research findings directly contradict the claims made about that time in the bible. In such a case the conflicting evidences have to be weighed against each other for plausibility.
There are different fields of scientific enquiry dealing with the subject matter: archaeology, history and theology. These fields have requirements for evidence that differ only in their details. From a historian's point of view finding one piece of evidence is very interesting but an undesirable status. Finding two pieces of evidence is more desirable. Sometimes you only have one have to make do. But if you have at least two pieces and the two pieces are in essential agreement and independent from another then a very important step in the direction of reliable evidence has been made. If some theologians are content with the single evidence they get from the bible that is another matter.
One example is the unsettled debate over the existence of the historical Jesus. If you are looking for two independent sources for his existence you cannot use the gospel of Mark and the Gospel of Lukas for that. Those are dependent from one another (although the exact relationship is still a hot matter of dispute). Finding a letter of Pilate detailing the circumstances of a very unusual execution of a jewish troublemaker and an Alexandrian account of a very unusual passover around 30 CE reported to them from their Jerusalem brethren in faith at roughly the same time and you have the perfect sensation.
So, it may be less surprising now, but: the bible is evidence. The bible does not get special treatment because it is the bible. But this is true either way. The bible is not discounted as evidence; but every piece of evidence is required to be treated the same.
It is literary history in some of its parts (we would all agree that the psalms are not?) with varying degrees of reliabiliy. One big problem is that the actual texts are written with a huge historical distance from the times depicted in them in the case of Davidic kingship and everything before.
While some argue that a few of the earliest texts in the Old Testament are probably written in the 8th century BCE some even argue that the core texts of the Old Testament weren't written before the exile and remained in flux, slowly solidifying into the received text until the 2nd century CE.
That means eyewitness accounts or those accounts of contemporaries are not extant for the time in question. As time progresses historical depictions get more and more distorted. Case in point: The hittites mentioned in the bible (one of the sins attributed to David was that he sent Uriah the Hittite to his certain death for want of his beautiful wife (2Sam), Solomon extracted tribute 2Chron)… were indeed regarded as literary invention.
Then the ruins of Hattusa were found and in there the vast archives of a once big empire. Triumph for biblical archaeology! "Atheist science doubted their existence as Bible hokum. Yet: The Hittites really did exist!" That is in a sense very true. But it is a double edged sword for biblical literalists and fundamentalists:
The biblical Hittites are not the Hittites.
The Hittites as depicted in the bible are indeed a military mighty enemy. Unfortunately this gets into conflict with the universally established timeline: The vast Hittite empire centered in Anatolia wholly disintegrated in the great bronze age collapse. What remained or re-emerged in its place and certainly so in the close neighbourhood to the Israelites were the so called Neo-Hittite (City-)States. Carrying forth the culture and traditions for some time. Much smaller in scale and far removed from the once powerful empire of kings like Hattusili and Suppiluliumas. Conflicts between those two entities – Israelite Davidic empire and Hittite empire – on that scale are off by a few centuries. On the other hand: conflict between a small Davidic fiefdom (very well may have been on the rise then) and a few local Neo-Hittite small powers are a very plausible match – yet again: not on the scale as depicted in the bible.
Since the different fields of archaeology, history and theology are differing in their details when researching the same place and time, it is consequently difficult to present an over-arching consensus. Most researchers of archaeology and history seem to converge on the view depicted below although there is no complete consensus: mainly that the account given of the entire united monarchy is less than a historical narrative depicting actual events accurately but more like a mythological construct from much later times using the methods of history writing to lay out a plan for the great future to come. Within the field of theology there is even more dissent since some participants in that discussion insist that it is simply all true or that the evidence from the Bible has greater credibility than anything else.
What follows are the counterpoints to the view that the King David existed exactly as depicted in the bible or in the consequence, that such a king even existed at all:
In "The Bible Unearthed: Archaeology's New Vision of Ancient Isreal and the Origin of Sacred Texts" Finkelstein lists clearly, why "Appendix D: Why the Traditional Archaeology of the Davidic and Solomonic Period Is Wrong"
The most important archaeological evidence used to link destruction
levels with the Davidic conquests was the decorated Philistine
pottery, which was dated by scholars from the beginning of the twelfth
century BCE until about 1000 BCE. The first strata that did not
contain this distinctive style were dated to the tenth century, that
is, to the time of the united monarchy.
But this dating was based entirely on biblical chronology and was thus a circular argument because the lower date for the levels with
this pottery was fixed according to the presumed era of the Davidic
conquests around 1000 BCE. In fact, there was no clear evidence for
the precise date of the transition from the Philistine style to later
Moreover, recent studies have revolutionized the dating of Philistine
pottery. In recent decades, many major sites have been excavated in
the southern coastal plain of Israel, the area of strong Egyptian
presence in the twelfth century BCE, and the region where the
Philistines settled. These sites included three of the cities
mentioned in the Bible as the hub of Philistine life – Ashdod,
Ashkelon, and Ekron (Tel Miqne) as well as several sites that served
as Egyptian forts. The latter disclosed information about the
Egypto-Canaanite material culture in the last decades of Egyptian
hegemony in Canaan.
His conclusion is that the Omride dynasty might have closer ties to the biblical record, while the information in there about the King David is extremely unreliable, if not fictional at all.
An even more extreme view currently hold is:
There is no evidence of a United Monarchy, no evidence of a capital in
Jerusalem or of any coherent, unified political force that dominated
western Palestine, let alone an empire of the size the legends
describe. We do not have evidence for the existence of kings named
Saul, David or Solomon; nor do we have evidence for any temple at
Jerusalem in this early period. What we do know of Israel and Judah of
the tenth century does not allow us to interpret this lack of evidence
as a gap in our knowledge and information about the past, a result
merely of the accidental nature of archeology. There is neither room
nor context, no artifact or archive that points to such historical
realities in Palestine's tenth century. One cannot speak historically
of a state without a population. Nor can one speak of a capital
without a town. Stories are not enough.
The linguistic and literary reality of the biblical tradition is
folkloristic in essence. The concept of a benei Israel … is a
reflection of no sociopolitical entity of the historical state of
Israel of the Assyrian period… (Thompson, Thomas L. (1992). Early History of the Israelite People: From the Written and Archeological Sources)
From a historian's viewpoint it seems necessary to add that still not all information in the bible is:
- that the Bible cannot be considered reliable evidence for what had happened in ancient Israel; and
- second, that "Israel" itself is a problematic subject for historical study
The authors and editors of Samuel and Chronicles did not aim to record history, but to promote David's reign as inevitable and desirable, and for this reason there is little about David that is concrete and undisputed.
Summing up the relatively current views of theology oriented scholars
and historians, between "no historical David at all" and almost
maximal "full historical account":
“All in all, skeptical views of
David have not substantially changed the way the history of the early
period of Israel's kingship is written. Perhaps, in the future, David
will disappear from histories in the same way the patriarchs, the
matriarchs, and the exodus have done. Presumably, then, an
archaeological or sociological reconstruction of conditions in
eleventh- and tenth-century Palestine that depicts life in the area
before urbanization and before undisputed evidence of a monarchy will
replace him, or perhaps the beginnings of Israel's "real" history will
again be pushed later in time, and the time of David will be seen as
another mythical origin story. For now, however, David is
overwhelmingly seen as a plausible and understandable character who
was an important link between Israel's tribal period and the
full-fledged kingdoms of Israel and Judah.” […] “Archaeology has not
provided us with significant written remains from the early tenth
century, but scholars still speculate that a tenth-century king could
have employed scribes that kept records used in the day-to-day
administration of commerce, military affairs, and other concerns of
the court. Thus, common portraits of David today are usually cautious,
neither fully minimal, in which David surely had no large empire or
bureaucracy (or cannot be said to have had one), nor fully maximal, in
which he conquered an impressive amount of territory and ruled a
unified kingdom from Jerusalem. They are, however, most often very
biblical." Excerpt From: Megan Bishop Moore;Brad E. Kelle. “Biblical
History and Israels Past: The Changing Study of the Bible and
History.” 2011, pp. 232–233, 289/90 (e)
That leaves the concrete questions of the OP at:
Being one of the first kings of the Israelite people?
-- Somewhat yes: the range being from maybe there actually was an important local chieftain to local king.
Conquering and settling Jerusalem (called Jebus before the conquest)?
-- Unclear and unlikely but not entirely impossible.
Conquering relatively vast territories and bringing the borders of the kingdom of Israel to their biggest extent ever?
-- Almost certainly not.
Actually being called David (or something resembling that name)
-- Considering the first three questions: somewhat yes, but insignificant in its consequences.