I think we can chalk up Hal’s defeat of two apparently superior fighters to dramatic license, but there’s some other stuff I wanted to comment on, in the hopes of getting a good discussion going on Shakespeare’s histories… 
But I don’t think we’re meant to see it that way at all! For reference, here’s what he actually says, first time he gets the stage to himself:
I know you all, and will awhile uphold
The unyoked humor of your idleness.
Yet herein will I imitate the sun,
Who doth permit the base contagious clouds
To smother up his beauty from the world,
That when he please again to be himself,
Being wanted, he may be more wondered at
By breaking through the foul and ugly mists
Of vapors that did seem to strangle him.
If all the year were playing holidays,
To sport would be as tedious as to work;
But when they seldom come, they wished-for come,
And nothing pleaseth but rare accidents.
So when this loose behavior I throw off
And pay the debt I never promised,
By how much better than my word I am,
By so much shall I falsify men’s hopes;
And like bright metal on a sullen ground,
My reformation, glittering o’er my fault,
Shall show more goodly and attract more eyes
Than that which hath no foil to set it off.
I’ll so offend to make offense a skill,
Redeeming time when men least think I will.
(1.2.189-211)
And of course we actually see him do all this, so I see no reason to say he’s lying to himself. It does color the way we see him over the rest of the tetralogy, though – Shakespeare even reminds us of it in Henry V. (It also creates some interesting choices for actors and directors.)
Incidentally, there is a niftly little scene in part 2, right after the death of Henry IV, where the Lord Chief Justice and some of the other noblemen are worrying about their prospects in the new reign, and the LCJ effectively says, “Well, now we’re all going to have to kiss Sir John Falstaff’s colossal behind…” 
Finagle, good call on divine justice. Though how one might interpret that with regards to the events at Shrewsbury is a bit of a thorny question (since it’s the forces of the usurping king that win). I think the strictly providentialist reading is that Hotspur the rebels are punished here for their rebellion, while the punishment of Henry IV’s usurpation is brought about by the spectacular incompetence of his grandson, Henry VI (who is, in the ironic way these things always seem to work, a genuinely virtuous man). I don’t think Shakespeare is as cut-and-dried as all that – I don’t think very many people do, especially nowadays
– but he’s very concerned with the question. (There’s a lot of writing out there that addresses it more thoroughly and eloquently than I can here.) It’s particularly explicit in Richard II.
wizard song – the proposal that Bolingbroke is the real hero of Richard II is intriguing, though I don’t agree with it. I suppose it depends on your thoughts on the questions I’ve mentioned above, and how you define “hero” (e.g. is Richard III the hero of his play? Though I suppose “antihero” is probably the best word here) but otoh I don’t think he commands enough of the audience’s interests to really be very heroic. The rest of what you said, though, I can pretty much agree with. “Henry V can only be said to have a passing relationship with the moral highground in his play” – well said. 
Though most of the actual dispatching of rebels in 2H4 is left to Prince John of Lancaster (and his dirty, underhanded tricks :p). Henry IV’s deathbed advice to his son is interesting, though: it amounts to the time-honored strategy of “Go start foreign wars so that everyone will forget about your dubious claim to the crown.” Works like a charm, too…