Hey, iondot. You can totally stay within the budget & be happy. My post about building the system should make clear that I know a lot of people who've stayed w/in their budget & I know they're as happy as me w/ their system. Just splurging on the gear I liked is probably just a testament to my lack of discipline more than anything!

On speakers, if you're going to go w/ the Audioengines, you should go for 5 if you'll be listening in a bigger room. Especially if you'll be listening to Arvo Part. I haven't A-B'd Audioengines w/ other bookshelf's or anything, but I am sure the PSB Alpha 1 or the Infinity Primus 162s will give you greater fidelity in rendering recordings. I heard the Infinity 162 speakers & they really are a steal at that price, as R. Reina mentions in his review. The bass is surprisingly robust, and the music sounded really good. At one point, when I started looking for a system, I was going to settle on the Infinity 162 with a decent amp, like a used Atoll. I really liked the Infinity speakers...

Amp-wise, I'm sure you can find something at Audiogon that will fit your budget... I hope jackfish chimes in here with his usual lists of amps/speakers that go with different budgets... but one of the options that I looked at was Harman Kardon 3485. I've heard almost all positive things about HK 3485. Great power, & you also get FM receiver. I also read in Head-Fi that it's got an amazing headphone-out, which is crucial to me, one of the reasons I went for the Leben, too. It's rare that an integrated comes with a good headphone-out in the first place, and a lot of people buy separate headphone amps. So for like $250 new (w/ free shipping on Amazon) I don't think you can lose... I think you may be able to find refurbished models for cheaper, too, if you're cool w/ that.

Nice musical selections. I would come over to listen to your system playing those (regardless of the stereo system!)

Tho AudioEngine 5 is not yet reviewed in Stereophile, but it has been positively reviewed by others (many others, as quoted on audioengineusa website). I would go for the 5 over the 2. Especially since you will presumably listen from places other than the chair in front of the computer. Remember it contains its own amp, so that really helps your budget.
I haven't compared it against the other mentioned good candidates, so I have no opinion there.
Anything mentioned so far will sound better than the Aiwa. How much better? I can't quantify that, sorry!
On the "budget creep" issue - I do think it's good to hear stuff at higher price points, since it helps you calibrate what to expect. It's entirely within your control - you may like more expensive stuff, say c'est la vie, and stick to your sensible budget. Or upgrade if you want. Up to you!