OK I have a good idea of what to recommend now. Let's start from the beginning of what you responded to.
"Intricate / sharp / edgy 1 - 2 - 3 - 4 - 5 smooth / forgiving / dark I'd say 2? Not too sure on the terminology here."
This usually is how you attribute a tweeter's sound. A tweeter that is scored a 1 or 2 are very detailed, but can potentially be TOO detailed, giving way to harshness. The opposite of that is a tweeter that is less detailed, but also less potential for harshness.
Given a "2", I'd say you need to look out for a tweeter that is pretty bright and intricate, but not over the top. You might want to go with a metal-dome tweeter, or a ring-radiator tweeter.
You also want a more up-front sound, more aggressive and more in-your-face. You want a speaker that handles pre-mixed / DJ sound better than natural, and you want it loud and clear. At this point I've got three strong ideas, based on price. But before I discuss that, let's cover your needs on the other issues.
This part is really, really important If a strong center image is something you want. In order to pull off a strong center image, you need digital signal processing (known as DSP). You need active time alignment control (either from your head unit, from a "smart amp", or from a stand-alone processor. You need to factor this in if you want any chance of getting a strong center image. Secondly, if you value accurate tonality, you will want some EQ control to get the frequency response in line. This is another job for DSP. If you want strong mid-bass impact, you need to invest in door woofers that can dig pretty low and stay with the system. You also need to deaden and seal those doors. You say you have 4 speakers which basically I think you are saying you don't already have tweeter pods up in the door. This isn't a big deal. Those might be the right places for a tweeter, or the A-pillar might be good as well. You should experiment before doing any final install work. Velcro or 2-sided sticky tape will let you fiddle with different positions. Here's where stage height is so important however: If you after a strong center image AND strong stage height, you need to run an active crossover and amplify each speaker individually after processing. Alternately, you can run a strong coaxial set (speakers with each other, not separate), and doing this you will lose your stage height, but you won't need to run active. You will still want time-delay left to right, but you won't need 4 channels worth. Lastly, don't even worry about those rear speakers. Stock, or near-stock speakers with low power will be more than you need. That center image you want literally depends on a front-stage focus.
OK, so let's move on to solutions given what we know. If you want a strong center image, and relatively high stage height, you need either a 4ch amp and a sub amp OR a strong 5ch/6ch amp, a processor somewhere in the system, and a good head unit. You won't be able to use a head unit with internal active crossovers because you need active control + rear speaker control. Option 1: Buy the Kenwood X4R amp which has internal DSP with all the right things (got one myself!), or Option 2: Buy a Audison BitOne.1, JBL MS8 (when it launches), or Rockford Fosgate 3sixty.2 (I've got two of those in my garage.) For speakers for the type of sound you are going for, Focal, Hertz, Boston Acoustics, Diamond Audio, JBL, or Kicker might be the right ticket. You could also go separate drivers like ceramic Tang-band tweets, and SEAS reed-cone woofers, among thousands of other unique combinations. Main focus here would be strong tweeters, and very strong mid-woofers. Budget? I've got it pegged right at $900-1200.
Here's another solution, if you give up stage height. A solid coaxial / braxial set of drivers up front, active time control on the head unit (or the Kenwood X4R used in a 3ch mode with that 3rd channel being a sub), and you're good to go. This will run significantly less, giving up stage height, and a bit of mid-bass punch. Cost? Right at 800-1000.
We'll go from there.