IRA contribution recharacterization of positions

I made a $7500 deductible IRA contribution for 2023 in January 2023, and have been recharacterizing positions that have done well in the past 6 months, rather than converting stocks that are still down from when I bought them (though I did a few of those too, and a lot of them in 2022).  Now my broker is telling me that when recharacterizing a contribution and having stocks (rather than cash) transferred to my Roth, I don't have to restrict myself to only shares bought after the contribution was made.  Is this true?  Can I recharacterize positions I bought last year, or even decades ago, as long as the total (I'm not sure how they'd calculate earnings) of the recharacterized amount is less than or equal to the contribution?

 

In some cases, I've had 500%+ gains in stocks over the years - so if I wanted to recharacterize say $3000 of my 2023 contribution, and have 10 shares of stock that's currently worth $290/share transferred to my Roth, with the remainder taken from cash, would they transfer all 10 shares plus $100 in cash?  Or would they look at the shares and see I paid $50 each for them, transfer all 10 shares for $500 in contributions, plus $2500 cash to make the $3000 recharacterization, and consider another $2400 to be nontaxable earnings?  I know that's an oversimplification, it depends on the value of the account before the recharacterization, but I'm just wondering how it works when 1.  Something has had a huge gain since the original contribution and 2.  The stocks you're asking to transfer to the Roth are ones that were bought before the contribution you're trying to recharacterize was even made.