This isn't tax-related, but you should be aware that you must have a Washington State nurse's license in order to practice nursing in Washington.
https://nursing.wa.gov/licensing/apply-license
Yes, rent you pay to your brother is taxable income to him.
And as @Opus 17 explained, you can't arbitrarily choose the location of your "tax home." Your tax home is determined by IRS rules and definitions.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
