Toyota Repair: 95 corolla electrical problems; a few of the exterior, as well as interior lights remain ON without keys in ignition..., stereo head unit, car stereo system

Question
QUESTION: PLEASE HELP! Hello, my car is possessed. Ok, its not, but my 1995 toyota corolla has major electrical issues (I think). I bought the car not too long ago, and am unsure of the vehicle history.
 I installed a car stereo system in my car, except for the actual stereo/head unit had previously been installed. I ran all the wire, used 4 gauge for power and ground, fuse, etc...the wiring was correct. I installed an MTX thunder 3401 amplifier also, two 12 inch audiobahn subwoofers. Everything was fine for about a month or so, then one night while driving (music on) the windshield wipers twitched/jolted/jumped a little. Upon exiting my car, I realized the automatic door lock button wasn't working. I tried to start the car and it was dead. I thought it was a bad battery, so I kept on getting jumps to start the car...eventually I checked the fuses. One of the fusible links located under the hood fuse box /junction block #1 was completely melted. I replaced it. The car still required a jump start. I checked the alternator...was barely reading btwn 11.75v-12.39v on the multimeter. I replaced the battery and alternator. Then a bunch of weird stuff started happening..one night while driving the car just dies and will not start with a jump. My friend comes and helps out, and decides to remove relays, flashers, fuses and not too sure if they were placed back correctly... While checking the obvious first, the starting system, I noticed a stripped wire touching the starter. I taped the wire with electrical tape and shortly after, the car just started by itself, no keys. I checked out the starter and checked the relays, flashers, fuses and noticed that the flasher (drivers side kickk panel) was in the incorrect spot. I fixed the entire car (fuses in correct location and whatever fuse it called for... that never happened again. I removed the stereo, head unit, amp amd subs. So here I am now, some of the cars lights remain on/lit even without keys in the ignition; interior instrument panel rear defogger indicator light, center dash lights (radio,clock, cig lighter), driver side automatic window light "auto" (on door handle). Exterior front corner/turn lights, rear tail lights. Turn signals, headlights, break lights do not work and the switch for the windshield wipers does not rotate (located to the right of the steering wheeL). Oh and I once again had to get a new battery, have yet to check the alternator again...I was going to 1. Check for a parasitic draw 2. Check fuses, learn how to read a wiring schematic and check for suspected shorts, bad grounds and continuity. 3.Check the ignition switch and everything around it. That's all I got so far. That's all I got so far. Please help!

ANSWER: Hello, I appreciate the detailed account of the problem.
As long as some of these lights and componencts are staying on there will be a draw on the battery which will draw the battery down so there is no need for a parsitic draw, we know this already.
There is no doubt that some wiring may shorted together or even grounded, wires shorting together usually result in some sort of feedback current from one circuit to another, a short to ground will allways result in blown fuses and dead circuits.
I think I have a Corolla wiring diagram so what I will ask you to do is the following:

Make sure any non factory wiring (stero/amp etc) are insulated or removed
Remove the key and make a list of all lights that are staying on both ninterior and exterior
Check all fuses and record the fuse designation for each blown fuse, for example "tail" fuse or "wiper" fuse
That's all I need for now.

---------- FOLLOW-UP ----------

QUESTION: Lights that stay on
Interior:
*Center (controls for the a/c, def, radio...)
*Defrost indicator light located on instrument cluster (instrument panel)
*Driver side armrest, 'auto' located on the power window button
Exterior:
*front side corner lights
*rear parking lights
*license plate

Also my blinkers don't work, nor does the switch for the windshield wipers. Maybe its the ignition switch?? Not the cylinder...also what do you mean by fuse designation for blown fuses?

Answer
All fuses are labeled on the covers both inside and under the hood, a bad ignition switch would not cause the lights to stay on, it sounds like there is a short circuit in the combination switch below the steering wheel, it gets power directly from the battery.