Yes, most functions work. You don't need any headers ofzo.
I'd qualify Gericom's comment a bit. There is some support for the lower level stuff in ASF (i.e peripherals). However, not everything is there and I don't think that much of the middleware in ASF is in the distro, although you may find some in the libraries folder for the various drivers that are supported.
In addition to the libraries you'll want to look in the folder hardware/arduino/sam/system/CMSIS/Device/ATMEL/sam3xa/include/component/ to see what has been compiled into the libsam_sam3x8e_gcc_rel.a library that sketches get linked with. If what you need isn't in there you can always bring it in.