I am beginning to think this problem is installation dependent.  The procedure you listed was one I had tried before and it still doesn't work.  Yet, clearly, it works for others.

I was able to start engines 3 & 4 by themselves using Shift-Ctrl-F4 and then F1 but as soon as I tried to start the engines 1 & 2, number 3 & 4 shut down and 1 & 2 didn't start.

Since you mention using F1, F2 & F3, is it safe to assume you are starting all 8 engines at the same time?