I don't quite understand question #1. Qualified HSA contributions are always excluded or deductible from your federal tax return. HSA contributions must be added back to taxable income for CA state income tax, but not federal, even if you live in CA.
If you both live full time in NY, then you are subject to NY tax laws and file an NY resident tax return, and you are not subject to CA state income tax laws, even if one of you works for a CA-based company. So you would get the full tax benefits of HSA contributions.
For question #2, you need to be aware of the full set of rules regarding HSAs. I will try to briefly cover the key points. Everything is here, https://www.irs.gov/pub/irs-pdf/p969.pdf
To make HSA contributions, you or your spouse must be covered by an eligible HDHP and no other disqualifying coverage. Let's suppose you have a family plan that covers your spouse, but you get a premium discount because her coverage is secondary to her own policy. That would disqualify her from making HSA contributions even if her primary coverage is a qualifying HDHP. Or, if you have an FSA with your insurance plan, that will disqualify your wife since your FSA can be used to pay for care for yourself, your dependents, or your spouse—therefore the FSA is "other coverage" that disqualifies her from making HSA contributions even if her primary coverage is a qualifying HDHP.
If your wife gets single HDHP coverage (only covers herself), her contribution limit is $3600. If she gets a family HDHP (that cover her and you, or her and children, or her and you and children) then her limit for 2021 is $7200. If she is covered by a qualifying plan as of December 1, 2021, she can contribute up to the maximum using the "last month rule" even though she was only covered for one month. However, the last month rule requires that she maintains a qualifying HDHP coverage for all of 2022, if she changes coverage, her 2021 contributions become retroactively ineligible and subject to a penalty.
You can contribute to an HSA in your own name if you are covered by an HDHP and no other disqualifying coverage. It can be a policy in your own name, or you as a spouse on your spouse's family policy. If you are covered by your spouse's family HDHP, you are deemed "covered" and you are eligible to make HSA contributions even if you are not the named policy holder. You might not have the option of an employer-sponsored account at that point, but you can open your own HSA at most banks for a small monthly fee.
(There are no joint HSAs, each account is owned by one specific person.)
You have the same overall family limit of $7200 regardless of your insurance coverage. In other words:
1. If your spouse has an individual HDHP and you have an individual HDHP, you can each contribute $3600.
2. If your spouse has a family HDHP that covers you, then your total contributions can be $7200, split any way you like.
3. If your spouse has and individual policy but you have non-qualifying coverage, your spouse can contribute $3600 and you can't contribute anything.
So whether you want to pay for one family HDHP, or two individual HDHPs, will depend on the premiums and other benefits of the plans.
You may not be able to switch plans mid-year, unless your spouse covers you on her plan (that would constitute a change that would allow you to drop your plan mid-year.). If you can't switch plans, your spouse could contribute $3600 for 2021 under the last month rule, and you would start contributing in 2022 when you are eligible.
