I am willing to install a Root CA and an Issuing CA with AD CS for my private environment (test purpose).
Do I need to obtain public OIDs (i.e. from IANA) or can I skip this requirement since my CAs are private within my domain?
For private or test environments you can use fictional object identifiers.
I suggest to obtain a free publically-registered OID from IANA: https://pen.iana.org/pen/PenApplication.page