Car Battery light, coolent light on, also automatically accelarate RPM goes up n down on its own

Hi, I have 91' Lexus LS400. This morning my car start perfect and took me to my service station to fill my A/C gas and I noticed that my radiator coolant cap was open (maybe I forgot to close it after filling coolant the other day)so I closed it and went to church without any problem, after I came back and start the car it RPM start to go up and down without me accelerating, and my steering got so hard I couldn't even turn it, so I put the car into drive and it came back to normal. I drive back to my house and I saw my battery light and coolant light, low oil pressure light came on but I kept driving and suddenly I saw all the lights on the dash went so low (not so bright) even the A/C blower went down and A/C compressor stop working but the blower start to blow air, after few minutes of driving car died so I jump the car and start to drive not even a minutes car stops again so I checked with a near by service shop for the fault and they said it was my Alternator so I replaced the alternator with a used one and we checked for the out put power it was 13.5 but the car lights not bright.

so we kept the car run for awhile and turn off and on again and the same problem came back (RPM up and down)when the car is on drive it goes but on P/R/N get the problem. When changing gear radio shutdown n come on again, some interior lights become bright and go dim some wired things happen.

I really don't know what is really going on. Any help would be great.