The expert here is @dmertz .
Recharacterization: I tend to suspect not, because that would require treating the contributions as taxable income and they would have to be added back to your W-2 income and subject to tax withholding by your employer. 401(k) plans, with or without the Roth option, are not IRAs, and are governed by different regulations, so even though they have the same overall purpose, they are very different in some other ways.
Conversion: This is allowed by law, the question is, is it allowed by the plan? Probably, but you have to ask. Most plans will not allow you to withdraw money as long as you are still employed by the plan sponsor. But a Roth conversion is not a withdrawal, it's a conversion, so it's allowed by the tax code and will probably be allowed by the fund manager. If you have a 401(k) account from a previous employer, you are definitely allowed to withdraw, rollover or covert those funds, and you could roll them over to the current 401(k) (which is usually allowed, ask first) or you could convert them to a private Roth IRA.
Conversions are reported for tax purposes as of the day they happen, you can't do a conversion in 2023 and call it part of the 2022 tax year. If you want to do a conversion from pre-tax 401(k) to a Roth option 401(k) in 2022, you will want to initiate that by December 20 or so, so it has a chance to "settle" before the end of the year, given the inevitable delays around the holidays.
