The answer to your questions are as follows. You must file all of your worldwide income to your resident state, and if you have income that is earned because you physically work in another state, the the income earned while you were present in the nonresident state must be filed on that state return. In this scenario, your resident state will give your a credit for taxes paid to another state.
If you were not physically present in Delaware (DE) while the money was earned then on your personal/individual return there would be nothing to file in DE. If this is true then you would file only your resident state of California (CA).
If you have a business that is situated in DE, then the business return must be filed in DE.
For the second question, if you have expenses as a self employed individual, then you must use TurboTax Self-employed to utilize any expenses for the business. Otherwise the business can reimburse you and then the business return can take the deduction for these expenses. If you are an employee and do not receive reimbursement from your employer, then these expenses are not deducible.
For more information about employee business expenses this link will be helpful: Can employees deduct job related expenses?
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