I jump start my wifes car at night with the cables backwards

Question edited for clarity, 'fuse', unless you light it 'fuze'.Question moved to sub category.

If the radio has an inline fuse, it will be within 82 of the plug at the rear. They are built into the harness. Unlikely because there are three fuses covering audio.

https://fuse-box.info/isuzu/isuzu-trooper-1992-2002-fuses-and-relays

C11 10 (AUDIO[ACC]) MIRROR Audio, Door mirror, Door mirror control SW, Door mirror folding SW, Digital clock, Speaker C12 20 CIGARETTE Cigarette lighter C13 10 ANTI THEFT Anti-theft controller C14 15 STOP A/T CONT Stoplight SW (w/o cruise control), Brake SW (w/cruise control), Rear wheel anti-lock brake controller, Stoplight, Transmission control module, Electronic brake control module, Trailer harness connector, Cruise control unit, A/T shift lock control unit, Lockup off relay C15 20 AUDIO[BJ] Horn, Horn relay, Horn SW, Hazard warning light SW, Flasher unit, Anti-theft horn, Hazard warning light, Audio C15 20 TELEPHONE Telephone C16 10 CLOCK[B] ROOM Digital clock, Audio, Dome light, Map light, Luggage room light, Courtesy light, Door SW (Front, rear, tail gate), Antenna, Anti-theft controller, Key remind SW, Seat belt, Key & light remind buzzer

Hello. It seems your Toyota's main electrical relay is 'shorted out'. The electrical accessories you stated (normally functions) whether the ignition switch is on or off; doesn't matter. It's, more and likely a relay connected, directly, at the battery. Also, check the 'fusible link'; also on the (+) battery cable. Good luck and keep me informed; if you want to.

Might need to check the main circuits for a blown fuse or fuseable link.
There is a fuse wire between the alternator and battery.

Well, something is on if the battery is dead. Pull the interior light fuse, and the clock fuse, disconnect the negative battery cable and put a 12 volt test light between the battery terminal and the cable...Likely it will light. Then pull fuses one at a time and observe if the light goes out. Keep disconnecting until you get the light to go out. Do not rule out the new alternator, stuff happens.

OK THE PROBLEM HERE IS IF YOU REMOVE THE MEMORY FUSE YOUR CLOCK RADIO AND INTERIOR LIGNT WILL NOT WORK. BUT IF YOU CHANGE IT(IF BROKE OR MALFUNCTIONAL) YOU WILL RESET CLOCK RADIO AND LIHT.
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KIA MOTORS COMPLAINT

did you see the smoke before or after you jumpstarted the car? if it was before the cables were hooked up incorrectly and with the switched polarity it can not only kill your battery but also computer components. if the smoke was after the jump start, depending on the car jumped, their alternator could have sent too much amperage to your battery causing an overcharge and you will need to replace it. A battery can turn lights and the radio on with low voltage but it takes at least 12.6 to turn your starter and start the car.

Your Vehicle has two fuse boxes one fuse box under the hood and one inside the vehicle. Check your owner’s manual and you should find the fuse panel at the side of driver’s dash console. Look at the fuse schematics for the fuse description radio then remove that fuse for a visual inspection. If it is good it could be an internal radio problem. However, If the battery jumper cables were hooked up incorrectly it may have damaged a battery cell or the regulator of the alternator and I would have her take it to NAPA, O’Reilly, or Auto Zone to have a free battery and charging system test.

sounds like you need to replace the radio

there are a few diodes in the electrical system that have been blown.  Some are in the dash and some are in the different control modules in the car.  Your insurance should pay for this problem.

your radio is staying on? you have a bad door switch.the bcm does not see door opening and thus does not turn radio off. most likely the drivers door.