I know that theoretically it is possible to authenticate a 4G simcard over a 5G core. Am I still possible to do it on Free5GC? If yes, which authentication method should I use for a 4G simcard connected to Free5GC? 5G_AKA or 5G_AKA_PRIME?
A 5G SIM card (or to be exact a UICC card with a 5G USIM application on it) ist mostly needed for features like privacy protection over the air interface (SUPI/SUCI). So if you want to use a 4G card, that is ok, but some 5G features may not work, as the file structure in the SIM card has been extended and the files are just not there.
on your question, 5G AKA or 5G AKA prime. I suggest to use the 5G AKA, the prime is intended for non-3GPP access.
just a small hint, when reading. A SIM card is actually a 2G card, while a UICC card containing a USIM application is used from 3G onwards. Spoken language is a bit sloppy on the terminology.