The doors should be able to take up to a 5.25" speaker. Pay attention to the depth clearance you will have from the window and mechanical components.
Not sure about the front corner lights on the 88 civic, might want to check in the civic forum for more details. I would assume they slide out unless youre missing a clip somewhere holding them in.
Revers lights, check your fuses. Also check inside the light plug make sure there is no corrosion and there is good contact. If you have a test light you can check to see if the wires to the reverse lights are recieving power. If not it could possibly be the contact sensor when the car is put into reverse not working properly.
Good Luck.