The DSP is actually like modern DSP as far as I can tell and was rather ahead of it's time as most were not even aware what they were 20 years ago, let alone offering one in a car from the factory.
It basically stages the speakers differently, by using time alignment or whatever. Mine had front/rear door speakers AND deck speakers. AFAIK, the F/R doors were controlled by the factory HU and the rear deck 6x9's were from a factory amp. w/o a DSP, the sound from the speakers might hit your ears at different times, assuming your fade/balance are set to 0/middle. So you'd basically ALWAYS hear the front doors prominently, then maybe the rear doors, and finally the decks, or maybe the decks sound more prominent than the rear doors, etc.. The DSP tries to fiddle with the sound so it sounds even or whatever, AFAIK. I know little about them tbh. They are also supposed to be able to affect the sound contour limitations of factory HU.
My '99 had an Alpine cassette deck... with the DSP, which was a rather odd feature to order in '99. I played with it a bit before pulling the stock HU, and from what I can tell, you can mimic the effect IF using multiple amplifiers. Ideally, you'd have say a separate amp for each door speaker or 2 amplifiers (one for left side of car, one for right side; this cuts down on cross-talk i.e. you pan balance all the way left, your right channel should be dead silent; cross-talk will allow some sounds to "bleed" over from left/right if I understand it correctly, so if you are listening to an older rock song where they panned vocals hard right, guitar solo hard left, and you are playing it loud enough, you might be able to hear sounds where they are not intended. Would be easier to hear on headphones).
Anyhow, assume an amp for each door speaker, and either a 2 channel amp or 2 more for each deck speaker. Turn off any amplifier cuts/freq. cuts, etc. for door speakers and adjust the gain to where each speaker can be heard from your seat, and use a HU high pass cut to protect them. Then mono out the rear deck vs. left/right stereo and add a separate cut and adjust their gain so they blend with the rest of the system.
I did something very similar to that, and to my ears, it sounded almost identical to the Alpine DSP from the factory for the one setting that made the rear deck more pronounced, but to be fair, I also had 4 tweeters top mounted near the headliner (1 above each seat area on a 4 door) and firing at the seats and I could hear everything.
DSP basically try to correct odd car layouts. I honestly think they are more or less hype from the fabled internet where person A reads person B has one and person B writes how great it is......
As far as your amp and the 8's, don't use the factor wiring. Chances are, your amp (if/when working) will pull more current than the factory wiring would allow. Your best bet is to buy one of those 4 or 6 gauge amp wiring kits, run your own 12v+ and ground directly to chassis (BARE METAL TO THE GROUND WIRE; paint, rust, etc. will cause issues), run the RCA cables with a high/low adapter and run a switch to the remote.
If it were me, splurge and buy a $100-150 HU with a CD player, USB, BT, etc. and at LEAST sub pre-outs as those HU usually offer bass cuts, a line level adjustment at the HU for the subs, etc. etc. Run the RCA to the amp, and run the remote 12v+ ON to the HU (can still add a simple toggle in between that if you want) and the HU will always power on the sub amp unless you switch it off. Also, strongly suggest the HU has a high-pass filter built-in. Set the HP filter to around 80-100hz. The factory speakers are actually OK (they have a Made in the USA stamping and might actually be Alpine) for use with aftermarket HU, but you MUST cut bass freq. to protect them. w/o the cut, they try and go as low as the HU allows, and they'll break up and die prematurely.
Sorry for the long post. There might be some inaccuracies with the DSP stuff (not my expertise), so read up more at good sources if you want to learn about it. Some people swear by them, but then again few people will chastise a several hundred dollar investment if it adds marginal improvements. And if they grew up listening to BPC stereos from the 90's, and never anything better, these DSP-systems might be like listening to a McIntosh for the 1st time after only hearing garbage home systems.