What is the ruling on inviting to and gathering for circumcision, what are the regulations for such a gathering, and did the Salaf (may Allah have mercy on them) have an opinion on this?

The invitation to a circumcision feast and other invitations that are not wedding banquets are recommended, and responding to them is not obligatory. This is the view of Malik, Shafi'i, and Abu Hanifa. They based their argument on the fact that the Sunnah (definition) explicitly mentioned responding to the wedding banquet (marriage) invitation only, unlike the circumcision invitation. Furthermore, marriage is encouraged to be announced publicly and attended by many people. As for the command to respond, it is understood as a recommendation, as it involves comforting the inviter and pleasing their heart.