When I want to check what was going on in a system,I use the "Play Movement Log Stepped" in that system. I can then see all movement in this system day after day. (You will have to use the Play Movement Log button to clear the stepped playback though, but it only shows all movement in a given system so it should be fast enough)
It is useful for recon in a system as well, order a ship to warp in an unknown system, and then to warp out as soon as possible. If there is no defence on the other side, you will have the information you need without any notice. (Unless you are unlucky and the enemy does look at the movement log in this system) Slinky, and doesn't tell the Evil Ones you are planning something.
I concur with PvK about staying clear from the "Play Movement Log for all ships" or something along these lines, as it does include every single system in the whole armada.
As for your other question Minipol, you can save the game during a turn even in simultaneous thanks to the Last patch. Save the game while doing your turn, and then load this savegame when you want to play again. You will resume your turn when you left it. (Only works with the latest patch, that is, 1.91)