Deductions & credits

You have some misunderstandings.  An HSA is owned by one and only one person.   See below for your specific questions.

It seems from your question that you are both enrolled in a qualifying HDHP.  You don't say if it is a family HFHP or an individual HDHP.  Possibly one of you has a single plan and the other has a family plan that includes your child.   As long as one spouse is enrolled in a family HDHP, then both spouses are considered eligible to make HSA contributions up to the family plan limit, even if the other spouse is not actually named as the policyholder on a family HDHP.  There is an overall family maximum of $7750 for 2023, so one spouse could contribute $7750 and the other spouse nothing, or each spouse can contribute $3850, or one spouse can contribute $1000 and the other spouse $6750.  Any split you like as long as the overall total is not more than $7750.

(There is a special rule if you or your spouse is age 55 or older, but I will ignore that for now.)

Therefore, you and your spouse are both eligible to make HSA contributions.  However, HSA accounts are owned by one person.  You can make contributions to your account and your spouse can make contributions to your spouse's account (if they open an account).  HSA funds cannot be moved, transferred or shared between accounts.  

One option is to enroll in the employer sponsored HSA by payroll deduction.  Another option is to open an HSA at any bank that offers them, and make tax deductible contributions out of pocket.  Instead of getting a reduced paycheck, you would claim the deductible contributions on your income tax return.  However, note that if you contribute by payroll deduction, you save an extra 7.65% on taxes, because the payroll reduction reduces your state, federal, and social security and medicare tax, but making out of pocket contributions only reduces your state and federal income tax.  Also, the workplace plan might have a lower monthly maintenance fee, because the fees might be subsidized by the employer.

So, you have several options:

1. You can enroll in payroll deductions and use the employer sponsored HSA, and contribute $7750 for 2023.  Your spouse does not have an HSA.

2. You can enroll in payroll deductions and use the employer sponsored HSA, and contribute some convenient amount less than $7750, while your spouse also enrolls in the employer sponsored HSA and contributes some convenient amount less than $7750, so long as your total combined contributions are not more than $7750.

3. You can enroll in payroll deductions and use the employer sponsored HSA, and contribute some convenient amount less than $7750, while your spouse opens an HSA at a local bank, and makes out of pocket contributions, so long as your total combined contributions are not more than $7750.

Any person with an HSA can use the funds to pay for qualifying care expenses of themselves, their spouse, or their dependents.  

1. Can a HDHP owner fund two HSA accounts but stay within the maximum family limit? i.e. can I fund my HSA as well as my wife’s HSA with $4150 each?

You can own more than one account (such as at different banks, although it seems like more trouble than it's worth) but your overall limit is $7750 if you are enrolled in a family HDHP and $3850 if you are enrolled in a single HDHP.  (The limits of $4150 and $8300 are for 2024.)   You can't fund your wife's HSA but your wife can fund it.  (This may seem like a minor point but it can be important.)

2. If I can’t fund two account directly, can I transfer funds from one HSA to another without paying taxes or exceeding limits? i.e. transfer $4150 from my HSA account to my wife’s HSA account

No. An HSA owned by one person.  You can't roll over, transfer or move funds from your HSA to someone else's, even if they are your spouse.

3. If I contribute the full $8300 from my paycheck, how can my wife “pay me back” for her pre-tax $4150 portion? What tax rate should we apply to the $4150 so her take-home-pay remains the same as if she would have contributed those $4150 from pre-tax money?

This is not a tax question.  You have chosen to manage your finances in a peculiar way, so whatever works for you.  Making HSA contributions at work will reduce your taxes by 7.65% for social security and medicare, by some percentage for federal income tax (12%, 22% or higher depending on your income) and by some amount of state income tax (3-13%, depending on your state and your income).