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1. I understand that I need to pay CA state tax for 3 days business trip in CA, but do I pay CA state tax for vacation in CA?  Yes, any days spent working IN CA would be income taxable to CA.  So if your business trip/vacation were all together, then the days you actually worked would be taxable.  If you were on vacation and not working, then those days would not be counted for anything.
2. How do I get the amount for the question "How much did you earn in California?"  The amount you earned in CA would be calculated by taking your daily pay multiplied by the number of days you worked in CA.  If you are salary, then you would divide your total salary for the year by the total number of days you worked during the year and then multiply that by the number of days you actually performed work in CA, even if your employer did not list CA as your work location. 
3. For the number of days mentioned above, I counted the number of nights I stayed in CA, including weekends nights, even I didn't work during weekends, is this correct? Should I exclude weekends nights? You would only count the actual DAYS you worked.  You would not count weekends or days you did not work. 
4. I had other business trips to CA in the rest of 2023 during which my work location has been changed to my home state on my payroll, so I don't need to provide these trips infos on CA state return, right? This does NOT change the taxability of the income.  If you were physically present performing a service (business trip) in the state of CA, whether or not your employer withheld taxes for CA or what they indicated your work location to be, you would be liable for taxes in CA.

However, depending on how much you earned, you may not have to file a CA return at all.  As they do have deductions and exemptions that may cause you to have $0 CA wages.  The best thing to do is walk through the CA return to see if you actually need to file or not.