Yes- I use the Sawyer pump spray on pants, shirts and socks. A must for hanging treestands in the summer and even early bow season in these parts.
No- it does not cause scent problems. Some folks will dilute the concentrate found in ag stores (I've done it) and the concentrate does have a "smell".
I air dry the clothes outside for a day or two and it's fine. Treating the tops of your socks is important here because of chiggers.
The extra movement from pulling ticks and scratching chiggers would spook more game than scent if you are playing the wind right.
Edit: When I occasionally use a floored tent in a buggy area I have sprayed the outside of the flooring but depending on the material, its absorption can vary.