#1. - it depends. This is because CA is likely still your domicile, BUT you might be considered a nonresident for 2022 (it depends on how long you were gone).
CA says, "Temporary and Transitory Absences from California
If you are domiciled in California and you worked outside of California for an uninterrupted period of at least 546 consecutive days under an employment contract, you are considered a nonresident. This provision also applies to the spouse/RDP who accompanies the employed individual during those 546 consecutive days. However, you will not qualify under this provision if you are present in California for a total of more than 45 days during any taxable year covered by the contract, or if you have income from stocks, bonds, notes, or other intangible property in excess of $200,000 for any taxable year covered by the contract. For more information, go to ftb.ca.gov/forms and get FTB Pub. 1031." Note, you can get FTB Pub 1031, but you have to give them your email address so they can email it to you (I have done that with other forms, so they actually will send it, I just don't know how long they will take at this time of year).
So, if you have not been out of California for 546 or more days, then you are still a resident of California because you have not changed your domicile. If you are still a resident of California, then you owe state tax.
Your domicile is where you get your mail, where you are registered to vote, where you own property, etc. As you can see, there are a number of factors that go into someplace being your domicile. It's the place you plan to return to. You stay a resident of a state, even when you move overseas, until you take positive steps to change your domicile - just sleeping someplace else doesn't do it.
So, while you were overseas, did you take any steps to change your residence? Did you apply for a residence permit in the new country? Did you buy property there? Did you take any steps with the local authorities to notify them that you were changing your residency? These are steps to take if you want to show that you are changing your domicile.
#2. No answer yet (maybe someone else will wander by).
#3. Yes, it appears that you may be eligible for the Foreign Earned Income Exclusion. Please see this IRS page for more information. Note the reference to the Interactive Assistant - this is an easy way to find out if you are eligible.
There is also the Foreign Tax Credit, but you have to actually pay foreign taxes to get this credit. Did you?
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