Desmond is not a Scoirish leprechaun. This is something that must be said. It may not be clear exactly whether he's Irish or Scottish, but that's the least of our worries with this mysterious "brother".
He likes to run (this being a literal sort of thing, unlike with a certain other Lost pup) and spent a lot of time training in L.A. for a solo race around the world. (This is, in fact, how he ended up on "the island" in the first place, but what little backstory he has is best left for later, perhaps.) He's therefore a very active person, and being cooped up in the hatch for all that time was /not/ his idea of marvellous fun.
However, he's rather a believer in things like karma and miracles, and he has a lot of faith -- if something's meant to happen, someone tells him to do something very important, he'll question it but he'll go along with it in the end. Entering the numbers into the computer, seeing Jack "in another life" -- that's just how the world works, as odd as it might be. He can do it all (though he wonders why) because he /knows/ there's no alternative. He keeps a vague sense of normality for himself, though, with a routine, music, tally of days on the wall and so on.
As much as he's willing to accept these things, he still feels threatened by entirely new changes that may be a danger to his established way of life -- he can get pretty intense in difficult situations, and determined to do what /he/ wants to do. This fits in with everything else: he'll go along with what he "knows" is right, but is reluctant for people to correct him -- even if he's also wondering, like Jack, whether it isn't all just a mind game...