See answers below:

If I do independent contractor work in CA and in WA, would that have an impact on how I do my taxes?  

Yes if you do business in multiple states you should also file state returns for each state.  Since WA does NOT have a state tax you will only need to file a CA return.  The CA return may be a resident or non-resident depending on where you live.

If in the future, I set up an LLC or sole proprietorship - what state would I set it up in if I do work in multiple states?  For example, I live in WA but do online and on-site work in CA for some projects. 

Since each state has it's pros and cons I would do some research to see which state your LLC will be more beneficial since it serves for both tax and legal purposes.  Please pay attention to the CA state rules as this state can be very aggressive when it comes to income being reported on your tax return.