"I took this job in February this year and had only physical stayed and worked for my employer in CA for 6 weeks, rest all was from Home office in Texas. "
The income you earned from those 6 weeks working in CA is taxable by CA. You will report that income by filing a 2020 non-resident CA tax return. Income from work actually (physically) performed in CA is CA-source income, and is taxable by CA regardless of your state of residence.
As long as you remain a legal resident of TX, the income you earn for your remote work performed in TX is not taxable by CA.
Once you move to CA and become a legal resident of CA, all your income will become taxable by CA.
As @Opus 17 stated, the business registration requirements of the state of Texas are beyond the scope of this forum, and in any case are irrelevant to your individual tax situation.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
