Should the full menu go in the invitation message?
No. One or two lines tease, the rest lives on the page. A full menu in chat commits you before replies are in, and menu changes then mean a resend. The message sells the dinner; the page holds it.
How do I ask about allergies without embarrassing anyone?
Put it on the reply form as an optional note. A guest types it where nobody else has to hear, and it comes back attached to their yes. Asking in the chat, even kindly, turns a small fact into a small speech.
How many days' notice does a dinner party need?
A week for a proper evening, two or three days for casual supper. Shorter works if the message is honest about it, and a dinner page helps because the time, address and menu all sit in one place guests can check late.
Is a plus-one fine at a dinner party?
With eight seats, every extra fork changes the plan, so decide before the message and let the form carry it. Each reply shows the plus-one room you actually allowed, which beats a guest guessing and asking in the group.
Can guests reply on a phone at the table of the invite?
That is the intended path. The link opens the page in the browser, the reply is a button press plus a note field, and it lands in your guest list while the soup is still warm.