What is the osmolarity of the processed liquid if the input side is fully loaded?
Just some educated guesses from me. I think it's safe to assume that the proccessed liquid will have an osmolarity higher than seawater to initiate and maintain forward osmosis as the output volume increases and dilutes the syrup. Depending on what reference you consult, seawater's osmolarity can vary from roughly 1,000-2,500 mOsm/L so the processed liquid's osmolarity probably falls roughly into this range, too. Just for reference, the body's normal osmolarity is roughly 300 mOsm/L.
If an insufficient amount of input material i.e. urine is used, is the processed solution of such a high osmolarity, that by drinking the solution could one become dehydrated?
In the FAQ for this product, it already advises users to sip the processed liquid, and to space out the sips, so even when fully loaded, the manufacturer already knows the processed liquid is concentrated enough to potentially cause dehydration, so I think it's safe to assume (there I go assuming again) that the answer to your question is "yes". At least, unlike sipping seawater and getting all that extra salt, the high osmolarity of the processed liquid is primarily from sugar and not salt, so at least your body can do something useful with the sugar.
What is the caloric content of the syrup?
The website lists 480 calories.