The shaking seen on the video is very normal for the fantails. It is also seen in other breeds including some Russian breeds.
However, for a Turkish Tumbler it is questionable in my opinion.
That brings me to your second question: do they have Syrian or Iraqi blood in them.
To me this is very much of a possibility but it could also be a different mix.
Since the live bird import laws in Australia are rather though the number of imports (as far as I know have been limited).
Maybe for this reason alone it is very though to find a pure bloodline.
Even the 100% birds tend to be mix of multiple reginoal sub-breeds.
This makes it very difficult to identify a bird based on looks alone.
I think, as you get babies from these birds and fly them you will have a better understanding of what is in the bloodline.
In any case, not all the birds in the video are from the same bloodline it looks like any way.
(Are the Browns and blacks from same family? What is the relations ship of the neck shaking checker with them?)
I would utilize the seperation by physical qualities method to dig a bit deaper into the bloodlines you have.
In the mean time keep an eye out for variations of beak sizes on the offspring.
Short (not shortface but shorter then what you see on the parents) and narrow/long (dove like) beaks will give away Arabic bloodlines.