Fixed gear bike: when I stop the bike completely I can move the pedals back and forth very slightly is this normal?
The chain should not be 'as tight as possible'. There should be some slack. If the chain is over-tensioned you will accelerate wear on it and the sprockets, and increase the possibility of snapping it.

There will be some pedal movement when the rear wheel is prevented from rotating, due to chain slack and the fact that the spaces between the sprocket teeth are larger than the chain rollers.