As the title shows I have a fixed gear bike, not sure if this is right though: even when the chain is tight as possible there will be what seems a loose spot and it makes it so that the pedals move back and forth slightly. For example, doing a track stand I can move the pedals back and forth slightly without moving the wheels, just wondering what might be the problem here, if anyone can help. It’s a Raleigh Falcon BTW so an old bike, that was converted to a fixie.
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.
Sounds like your chainrings, cogs or cranks are not machined to the tightest possible tolerances. Maybe the chainring is not perfectly round, or maybe the spider in the cranks the chainring attaches to is very slightly off-center. Usually it doesn't matter at all, even though it does cause the tighter spots you described, thus allowing a little bit of play in the loose spots.
Also, since it's a conversion, I'm going to assume you didn't use all-new drivetrain components. Any amount of wear on the chainring will cause this kind of play.
Then again, the issue you described is not really an issue but one of the inherent features of proper chain tension. I personally prefer even a bit more of that play in the chain as it gives me a clear feedback when my spinning technique starts getting wonky. As to this question, I suggest you re-check your chain tension an possibly loosen it up a bit.