This is pretty spoiler-laden, so if you don't mind finding out much of the Jedi Knight story without playing the games, feel free to take a look.
In JK, Kyle learns that his father was slain by the dark Jedi Jerec. He later has a vision in his sleep of an old friend of his father's, a Jedi named Rhan who was also killed by Jerec. Rhan tells Kyle he must protect a "Valley of the Jedi," an ancient battlefield that pulsates with the Force. Rhan finishes by saying that Kyle's destiny is to protect the Valley of the Jedi and learn the ways of the Force.
Kyle travels to his father's home on Sulon in order to find out more information. When he gets there, he discovers a lightsaber that once belonged to Rhan. This is Kyle's green saber in JK.
About halfway through the game, Kyle's lightsaber is destroyed. He takes another lightsaber that belonged to the former dark Jedi Yun, one of Jerec's seven dark Jedi. Kyle saved Yun during a lightsaber duel; when Kyle had the chance to kill Yun, he showed compassion instead. Yun reformed his evil ways by giving up his own life when Kyle was about to be killed. (This is assuming you choose the Light side in JK. The Dark side version of the game makes for a completely different story)
As for the Moldy Crow, it unfortunately was destroyed in a crash landing when Kyle escaped from a crashing Imperial starship. This happens right before he get's Yun's lightsaber.
The events that happen between Jedi Knight and Mysteries of the Sith are a little confusing. Kyle's second lightsaber in JK is yellow, but he has an orange saber in MotS when he succumbs temporarily to the Dark side. I'm not sure whether Kyle built a lightsaber of his own or if the orange saber is supposed to be Yun's yellow one. (maybe someone can clear this bit up, as I've never played MotS) In any case, it seems Kyle built another lightsaber right after MotS. (his blue one he uses in JO and JA) The most plausible theory I can give for his blue saber is this:
Kyle may have been one of Luke Skywalker's twelve original Jedi students in his academy on Yavin IV. (read the Jedi Academy trilogy books to find out more) Kyle built his blue saber because it was part of his Jedi training. However, the memory of his dark past in MotS haunted him. (he also almost fell to the Dark side in JK) He gave his saber to Luke and vowed that he would abandon his connection with the Force for fear of falling to the Dark side again and returned to his gun-toting mercenary ways. (this last part I am certain of) However, events in JO leave Kyle with no choice but to return to ways he swore he would never go back to.
*EDIT* Oops, looks like I took too long writing it up. Well, between the two of us you know quite a bit now!