can you take your dog on the park shuttle? Is the park trails off limits to dogs even on leash like Yosemite?

I doubt you can take your dog on the shuttle, but I don't know.
National Parks are definitely wild-animal-friendly and not pet-friendly.
From the 'pets' info page
>> For you and your neighbors to see and appreciate wildlife, please note that pets are not permitted on any of the trails in Sequoia and Kings Canyon National Parks.
In campgrounds and picnic areas, pets must be kept on a leash at all times. The leash must be less than 6 feet (1.8 meters) long. For your pet's safety, please don't leave pets in hot cars. Pets cannot be left tied and unattended at any time.
In the National Forest, pets are allowed on trails. <<
nps.gov/seki/planyourvisit/pets.htm
That page has a map that shows the boundaries of the National Forest and the National Park.