Regret + -ing form looks back at the past - at something that one is sorry that one did. I regret leaving school at 16 - it was a big mistake. Regret + infinitive is used mostly in announcements of bad news. We regret to inform passengers that the 14.50 train is one hour late. We regret to say that we are unable to help you.
Yes, the answer is "Do you regretgoing to the party?," not "Do you regret to go to the party?" When there is or may be regret for something that has already occurred, we always use either a (finite) "that"-clause or a (nonfinite) -ing clause. "Do you regret going to the party?" means the same thing as "Do you regret that you went to the party?"
The formal use of "regret to VP" is confined to verbs of speaking. "Regret to" is almost invariably followed by verbs like "inform," "tell," "say," "ask," "report," etc., where the clause following that verb does the informing. I think a sentence like "I regret to tell you this" may be analyzed as an abbreviated version of "I regret needing to tell you this," which contains the -ing complement after all.
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