Right up against the budget, but no compromises here. Should sound awesome. Rear speakers are superfluous.
$300 Front
$0 rear- keep current pioneers
$235 Sub and box
$220 5 channel amp
$60 wire kit
$160 sound deadening
If you look around hard enough, you could save some money buying welding wire vs a amplifier kit, but probably not much after inline fuses and spade connectors and such.
Front components are a plus, yes.
Rear, leave alone unless there is something wrong with the ones you have
A million ways to go on amps, but I went for simplicity with a single amplifier doing everything. I would bridge channels 1-4 into your front components and run the rears off the deck.
Dayton UM is my fav metal sub of all time! Diggs really deep for a sealed sub, and never sounds sloppy.
Dynamat is a big yes, it's really the only way to get that tight mid-bass out of a leaky door "speaker box", floors cut road noise, rear deck can help out midbass for the rear. Trunk kills rattle, etc.
Assuming you can't fit component speakers...
Order your head unit by phone to make sure you get any controls and harnesses you may need. Forget the rear speakers, bridge channels 3&4 into the subwoofer. Don't forget to seal and deaden doors