You always have to put the bass in a box, even if it's in a case. Shipping companies most likely won't accept it, and trust me, if they did, it would show up damaged. Atleast with scratches galore.
I use UPS 100%. Usually I have a box lying around from the last time I bought a bass (I tend to buy a bass, save the box, and wouldn't ya know, I sell the bass later and still have the box)

Other than that, I've either had certain shipping outlets make them for me, or I've gone to music stores and ask if they have any, which they usually do.
*edit* Forgot your last question. It varies on the price. It depends on how heavy the package is and where you're having shipped to. If you have the weight, package dimensions and the zip code and city of where you'd be shipping it to, you can go to
www.ups.com and get the rates. The most I think it ever cost me was $50 (from MA to CA)