Well, I'm bored, and very awake, so here goes, just because you seem to appreciate the advice...
For one You've got 6x8's up front which can make things tricky if you're wanting to get into something nice. The added bonus, however is that they've got alot more surface area than 6.5's so you'll get more low-end. At $300, you're really sort of hurting for what you'll be able to do. Any aftermarket deck really won't have any more power as the stock ones have been measured at 17w (generally more than an aftermarket deck would provide)
I'd reccomend to get a decent set of front speakers, and leave the 6x9's alone. infact, try fading them out a bit. And go for a small 2channel amp to push thenew front speakers with (again, don't worry about the rears, just give your front speakers some juice, they'll really come to life) Try this on a sound board if you get a chance. Take that cd you made (Excellent idea, BTW, that's the best thing you can do, use music that you're familiar with and enjoy) Find some 6x8's (or 6.5's if the 6x8's selection is lacking, you can make a baffle fairly easy) Anyway, find some you like the sound of Just powered off the deck at first, no sub, just deck and speakers. Then switch to just a smallish (35-50w RMS not max) amp powering those speakers, and see what you think. then try some of the others you may have previously passed up as well. 5x7 is the same thing as 6x8 don't let hype make you think different. Also, don't believe that more "ways" makes for a better speaker. generally a coaxile is plenty. The highest end systems generally only have one midbass and one tweeter per side. I personally for simple setups like the sound of JBL GTO and Infinity refrence 6x8's Don't think they need to be the most expensive, as spendier ones usually just require more power. my Infinity Kappa's don't sound good until they've got at least 70w going to 'em. With the amp, a simple Alpine, or eclipse would prolly be your best bets. Both are known to be nice and clean even with their inexpensive units. According to crutchfield, you could get an alpine 2x50w amp for 129, and a set of decent 6x8's for about $100. That leaves you money for shipping and amp-installation goodies. We can walk you through the amp install, it shouldn't take much.
That should give you a good starting point, but I highly reccomend you go listen to them and see for yourself if it's something you'd like. The nice thing to this is you can always add a sub later on. But if good clean music is what you want, nothing beats having an amp on your main front speakers. I know it's kind of against what everyone else seems to do, but this's a way to get your stuff sounding good. The bass won't rumble you at all, but you'll get good, loud, clean full-range music. In fact you likely won't even realize how loud it really is because it's not all distorted and cruddy sounding.
If you were more of an audiophile, I'd say drop the 3bills on a decent set of components and see if you can maybe scrounge up a bit more for an amp (actually the same thing low-end Alpine or Eclipse 2x50ish w RMS)
Components are superior generally inthat their crossovers are better meaning the mid and tweeter are more cleanly seperated whereas with a coaxile, generally the mid will play all notes, and the tweeter will have a very shallow sloped "cap" to block the lows which realistically only just kind of slowly tapers them off. With nice crossovers, you get a fairly sharp cutoff at say 4000hz for both the mid and tweeter so they don't overlap eachother. If you're getting excited about that, keep in mind that most low end "components" really are nothing better than a coaxile just pulled apart, meaning they don't have fully functional crossovers. Generally you'd need to spend about $200
As far as your settings preferances, generally that would suggest more than anythign else that you've got hardware shortcomings in your high and low end, both of which a nicer set of speakers with real tweeters and some power going to them will help. Ideally your settings should be flat and you should get the full range of any type of music you play reguardless of type.
A couple other notes. SONY SUCKS... Honestly, they make cool-looking stuff, but it just doesn't perform as well as anything else I'd be skeptical of this myself had I not had a couple experiemces namely when my stereo got swiped once a friend graciously loaned me his sony, and while it fileld the void, I NEVER liked it. it just wasn't right and I don't know why. The speakers speak for themselves when you listen to them. That may be a part of that "popping" sound you speak of. I'd also take that to be clipping meaning the gain is set too high, and he's trying to push something in the system harder than it wants to go (likely the amp) Another thing maybe that his crossover isn't set properly. General guideline is to set your sub-amp's crossover at about 80hz and have the rest of your speakers play above that. Natureally with smaller speakers there'll be a "gap" as 4's will really only play down to 150 or so with any kind of accuracy. This gap might be alot of what you're missing from the Bass drum that makes it sound ugly to you.
A note on the front-heavy "soundstage" we'll all throw at you. Basically it's a STEREO, not a quadrio. Decks cannot determine any difference front to rear aside from where you want to fade things. Now, for clean non-muddied sound. 6x9's are EVIL. they sit up on your rear deck where they reflect RIGHT off the window, and throw all sorts of noise at the back of your head. The worst part is because they're farther from you than your front speakers, the sound arrives at your ears at a slightly different time which confuses your senses, and like I said, adds noise. YOU don't got to a concert and turn your back to the stage, therefor, ideally you want your sound to "appear" as though it's in front of you, and strongly at that. Rear speakers do have their place, but most of us in here prettymuch don't use 'em. I've got 75w going to my door speaks (soon to be 120ish at a set of focals) and I actually had nicer amped speakers in the back, Wound up reverting to my factory speakers powered off of my deck because it sounds better to me that way. less noise, and mostly no tweeter (the highs behind your head are the piercing confusing ones)
Perhaps someone in here lives close to you and they'll let you experience what a stereo should sound like. If not, venture out. Ask questions, try to experience as much as possible 'cause in the end it's all about what soudns good to YOU, you're obviosuly not trying to impress judges. You're in the right place if you're willing to learn. Caught me in a good mood, sometimes ppl in here can be viscious, and they'll just bark at you to run searches (a good idea as well BTW) *whew* okay if i missed something or you need to know more. Ask away. These guys have helped me WAY more than I could ever express, I like being able to give back.
