You would need to keep the cat out of the are while you are spraying Tempo 20 WP, and until the product completely dries.
Tempo 20 WP has actually been discontinued by the manufacturer. As an alternative to that product you can use Martin’s Permethrin around horses and stalls.
Tempo 20 WP was discontinued many years ago. We do have Tempo SC Ultra or Tempo Ultra WP which can be used in an around homes.
Unless the chickens are consuming the Tempo 20 WP directly there should be no effect on the bird or their eggs. Once the product has dried, it only has a toxicity level of that to kill something as small as an insect. For there to be even a chance of your chicken having any effect from the tempo on insects, they would have to eat at least their weight worth in insects that have died from it.
The amount of product per gallon would depend on what you are treating and where you will be treating. Tempo 20 WP has been discontinued by the manufacturer. Please refer to the product label for more detailed information.