You ALWAYS pay state income tax in the state where you work it that state has a state income tax.
Most states with an income tax also tax residents, but usually have a credit for taxes paid to the state where you work.
Yes you do. You pay in the state where you earn the income and where you live. You would get a credit in your home state for tax paid to another state except if your home state had no income tax. Like Nevada.
If you live or work in California then you must file a California state income tax return and you must pay state income tax somewhere. If you pay income tax to another state, then you might not have to pay California (but you would still have to file) or you might have to pay California as well. If you don't pay anything to the other state, then you would definitely need to pay California.