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Grandpa Kim said:
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"Simultaneous" means simultaneous. Simultaneous movement happens over a period of 30 "days" in the month that one turn represents. Ships get to move once every 30/speed days. A ship with 6 speed, for example, will move on days 5, 10, 15, 20, 25, and 30. I'm not sure how the game handles rounding. All ship movement on any given day happens all at once and combat only happens after all the movement for the day is done. Note that "seek after" orders will seek after the location of the target at the start of the day. This makes it much easier to flee from an enemy fleet if you happen to be moving at exactly the same speed as your pursuer, and makes it possible for two fleets with the same speed to constantly "pass in the night", switching places all turn long while both seeking after each other.
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Not quite right concerning the seek order. If player one is seeking player two, he is sunk. He will move onto player two's fleet and then player two's fleet moves away. If player two is seeking player one, then player one will move and player two will move onto him (assuming he is still in an adjacent sector). In theory all players move simultaneously. In fact, the program handles everything one at a time. Welcome to the digital world. 
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I have seen the mechanism I outlined regarding seeking in action several times. I assure you it is correct. In test games and one actual single player game, I have observed fleets with the same speed and orders to seek after each other exchange places without combat each time they got to move. In a current PBW game I am in, a fleet I sent on a riot run was able to evade all pursuing forces with ease simply because all ships involved moved at speed 10, and every time the pursuing fleet moved to attack my fleet, my fleet moved out of the sector at the same time and no combat occurred. Note that I am player 11 of 11 in that game, and many times the sector I was moving to was also adjacent to the starting position of the pursuing fleet.
Edit: That's strange. I just tested it, and it worked as you described, but I remember very clearly seeing two fleets behave as I described in a single player game. Maybe it only works like that if the AI is controlling one fleet.
Edit2: After further testing, I have established that which player can flee with impunity is determined not by player number, but by order of construction of the ships involved. The player whose earliest-built ship in the fleet was built last gets to move second. Actually, it depends on the internal ID number of the ship, but as long as no ships have been destroyed it makes no difference - not that that situation is likely to last in a real game.