How much does a 1.6 Nagado Daiko Keyaki weigh, and what would it cost to send to California?

Taiko is made of natural wood and the weight depends on each item even though it is same product. But, 1.6 Nagado Daiko Keyaki would weigh 20kg to 30kg (about 44lb to 66lb). The most heaviest one for 1.6 Keyaki Nagado Daiko I've ever seen was 30kg. 

I'm interested in purchasing a drum. I'm assuming the bigger the drum (the height, and width) the deeper the sound? Is this a correct assumption?

Yes. Bigger drum makes deeper sounds. Hard wood like keyaki (Japanese zelkova) also makes the sound deeper.