"The contribution is immediately converted, thus a 1099-R is issued."

If the contribution is for 2023, the 1099-R will be from 2024 with code P = previous year.

"Does it get entered on the traditional IRA deduction screen, and then on the 1099-R screen"

Yes.

"(deductible and taxable)"

No. It should be nondeductible, so that the conversion is nontaxable.

"or is there a method to enter it so that it is non-deductible and non-taxable? Does the code on the 1099-R dicate how this is all entered?"

Yes.

Also, there should be no other Trad IRA, SIMPLE IRA or SEP IRA funds. The "backdoor" reference is for nondeductible funds that are converted so quickly, there is no even any growth, gain or earnings on it. It's all Basis. Otherwise, your taxpayer risks a pro rata taxable conversion.

The 1099 worksheet will trigger Form 8606.