Is Postfix capable of checking DKIM and SPF records on incoming email
and adding headers based upon its findings?
Not the postfix code itself, but postfix supports milters which do this.
My google searches have only returned results on how to do DKIM
signing on outbound email and not how to verify them on inbound emails.
The milter that does the signing usually will also verify signatures on
incoming, for example is OpenDKIM.
there are, as well, milters that verify incoming, but do not sign outgoing.
i typically use one with my postfix setup,
https://github.com/fastmail/authentication_milter