Unfortunately, one of the downsides of CA is the $800 annual franchise tax =[. Can't get around it even if you incorporate somewhere else.
Good news is, if you're an Inc you can waive the first year. LLCs can't though.
Yes
If you're doing biz in CA, I'm pretty sure that CA still collects corporate income tax. (I forget if CA still has a unitary corporate tax, but it definitely wants taxes on in-state profits.)