He's Shirou from the future. Does he not remember the events of this war?
In the version of the war that Archer experienced as a Master, Rin summoned someone else as a servant (still an Archer-class Servant, but we're never told who). Archer has no experience of having met his future-self.
I forget where - maybe it's some kind of side material or maybe it's Last Episode or something - but we are explicitly told that Archer vividly remembers having summoned Saber when he went through the war (this is something that happens in every timeline, because of Avalon). Something to the effect of "even were he to die a thousand times, he would never forgot Saber's radiance" or some similar schlock.
The thing Archer pulls at the beginning where he claims not to remember who he is is initially because Rin genuinely did mess up the summoning, but he regains his memory relatively quickly and maintains the facade of memory loss for some time afterwards.
How did Shirou die?
Archer explicitly mentions in episode 19 or 20 that he was scapegoated by one of the people he saved, and was executed shortly thereafter.
There is a scene in which a giant glowing ball absorbs Shirou (like some sort of pact), so I assume that's when he became a Heroic Spirit too. Did he die at that point? (so he killed himself)
Yes, that's when Archer (then-Shirou) "contracted with the world" (to use the parlance) and became a Counter Guardian. He only died later.
So Archer wants to kill Shirou because he regrets his decision of becoming a Heroic Spirit? Why does he regret it? Because Kiritsugu was right? (that you have to kill people to save others).
Archer regrets having become a Heroic Spirit because he did so by becoming a Counter Guardian (as opposed to by becoming an actual Hero). What this means is that he gets dispatched to situations where mankind is destroying itself. He stops mankind from destroying itself by killing the people who are the cause, generally after they've already done their damage.
Implicitly, this does mean that Kiritsugu was right, though I don't think that Archer is specifically bothered by the mere fact that Kiritsugu was right.
Why does Archer have Rho Aias from the Trojan War?
This is somewhat mysterious. The implication is that Archer must have been dispatched to the Trojan War at some point (in his capacity as a Counter Guardian), where he must have seen Ajax using his shield and copied it.
Is this Archer the same Archer from Fate Stay/Night? In other words, is the Archer from the original series also Shirou from the future?
By "Archer from the original series", do you mean "Archer from the 2006 DEEN anime"? If so, yes - that Archer is also Shirou from the future.