score:6
According to the schedule, the only train leaving the San Francisco Bay area for Los Angeles is the daily Coast Starlight. It is indeed a night train, but the overnight portion of the southbound journey covers southern Oregon and northern California. It leaves Oakland at 8:50 in the morning and arrives in Los Angeles at 9:00 in the evening. Therefore:
Are there any trains available from SFO on Friday night which reaches LA on Saturday morning and then from LA to SFO on Sunday night?
No.
The schedule linked above is the Pacific Surfliner schedule, because it covers more service, including several "thruway" services, operated by bus, which are denoted with a bus icon and shaded green in the timetable. The Coast Starlight schedule may also be of interest, but doesn't help in your quest for a night rail service between San Francisco and Los Angeles.
Upvote:2
I know you asked for trains, ruling out buses already, but there is an overnight bus service that will take you from SFO and drop you off in Santa Monica. That option is www.ridecabin.com. A round trip next weekend would cost about $230, a little more than most airplanes, but still less than the train phoog mentioned. It's possibly more comfortable too, considering the lack of comfort on airplanes these days. Friday and Sunday are their most expensive since most people want to do the same thing as you.
The only other economical option that I know of is to fly.
I don't work for ridecabin, and I have never tried them before. I did read an article about it recently that I remembered when I saw your question.