Ball screws have undetectably small backlash.
Depends on your application if it is 'undetectable'. My last machining centre had very expensive ball-screws, still had backlash compensation. For machining PCB's (basically no cutting force), the OP could use "rubberbands" to preload the lead-screws.
And your upright for the Z axis is gonna flex like a rubber band. Its the equavilent of cranking the quill out all the way on a bridgeport mill.
I doubt it in this application. Whilst I wouldn't to do anything vaguely heavy (machining) with the OP's mill as the neck doesn't look too rigid, I can't see why it would affect PCB milling.
How are you ensuring that the spindle is the same distance from every point on whatever bed you install? That is, how did you ensure the z-axis is perfectly orthogonal to the xy motion? I can't see clearly in the picture how you can adjust the alignment.
Quite simple, use the mill to machine it's own bed flat, or at the very least get it to machine the sacrificial slab that the PCB is going to sit on.