1) If you consider Texas to be your permanent home, then you would not need to file a Texas tax return because Texas has no state income tax. However, if you have income that is sourced in California, you will need to file a California income tax return.
2) It depends. If you are an employee of the California company, you cannot claim a home office deduction. If you are a contractor for the company, though, you may be able to claim a deduction for the business use of your home.
3) Again, if you are an employee and you incur travel required by the company that is not reimbursed, you cannot deduct those expenses. As a contractor, some travel expenses may be deductible. See IRS Tax Topic 511 - Business Travel Expenses for a discussion of what is deductible for someone who has two places of business.
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