Within Apple's certificates page, if I select a payment processing certificate and then download the .cer file, can this .cer file be safely distributed to third parties without any security concerns? I am unsure of the specific purpose of this certificate file - but if this .cer file is all the information a third party has, can they do anything malicious with it?
The certificate is effectively a signed public key. It's used validate signatures from your private key, the fact the certificate is also signed by Apple attests to your status within their ecosystem.
You can distribute it without concern, in fact, it will be distributed to anyone wishing to verify you in this context.
.cer files can contain private key material, but shouldn't in this example.