My 1991 Lexas LS400: All of the dash icons light up and radio goes on/off (at the same time) when applying the brake and/or while the `car has stopped. Had new battery checked, the battery is good. Recently had new alternator installed, alternator is good and replaced a bad alternator fuse. I was told it might still be the alternator, although I already had it checked or, it could be electrical. What other possibilities could there be?

I've seen this before, the problem ended up being the brake light bulb lead contacts touching each other on the dual filament bulbs allowing back feed voltage to travel through the park/dimmer switch circuit causing the lights to activate while pressing the brake pedal. I would pull all the brake light bulbs out and look at the back of each one, if you see nothing unusual, I would still replace them all only because it could be internal to the bulbs as well. Let me know if this fixes your problem.
I hope this info helps!~subyguru~