Are there any risks of publicizing a Lightning network invoice? What are the perils of publicizing a LN invoice either before or after its payment?
From what I understand, after its payment the second payment of the invoice cannot be done. Other than the fact that the invoice was already paid, what could the third parties learn from a lightning invoice?
Keeping aside social-engineering impersonations:

Can the safety of the funds by affected any any information published in the invoice?
In what way publishing such an invoice may affect privacy of the transacting parties?
Could the payer prove (cryptographically) that they were the one that paid the invoice?
Does it affect -most importantly- the safety of the funds?


It does not. Funds are not affected by publishing a LN invoice.


  Does it negatively affect the privacy or something else?


If a lightning invoice is published, 3rd parties may learn some semi-private information about the payer and payee. First and foremost, they learn the amount that was paid and to which node that amount was paid. Since an invoice can contain a short description, they may also learn what the payment was for by reading the description.

LN invoices also may contain specific routing information, particularly for private nodes. Revealing a LN invoice may also reveal a private node's existence, public key, and a channel id of that node.


  Could the actual payer prove (cryptographically) that they were the one that paid the invoice?


Anyone in the route (including the payer) can prove that they were in the route by revealing the payment preimage. I'm not sure whether the payer can prove that he was the payer, but he can prove that he was involved in the payment as either a forwarder or as the payer.

However if the payer is involved in a dispute with the payee, he can prove that he paid by presenting the payment preimage.