The Grip Claw is an item, so no. Baton Pass only passes stat changes. You can put the Grip Claw on the Pokemon you will be Baton Passing to, however.
Not most trainers, but you can find trainers every day in the Nimbassa City stadiums, you can battle trainers on the Ferris Wheel also in Nimbassa City, in Opelucid city there is a house in which you can do a Triple/Rotation battle every day, and in Pokemon Black there will be trainers that you can battle in Black City. Also, you can rebattle the Elite Four as much as you want and battle the rich boy in the Undella Town villa every day.