Does an aircraft landing on a carrier increase the carrier weight before it touches down?
No. Weight is defined as the effect of gravity on a body. Until the aircraft becomes part of the body of the carrier, it has no effect on the weight of the carrier.

I think you are confusing force with weight. For example, if wind shear drives wind against the top deck, this does not make the carrier have more weight, but it does cause the carrier to move down. Likewise any gust from an aircraft may apply a force to the carrier, but does not increase its weight. You might consider that force to increase the "apparent weight" of the carrier, but it cannot be said to increase the weight of the carrier itself.

Note also that the downward wind force of a landing fighter is relatively slight. For example, a person can easily walk under rotating helicopter blades and still stand up and walk. The force is probably only around 1-2 pounds per square inch, maybe 5,000-15,000 pounds total. For an aircraft it will be even less.