1. Note that you might say, equivalently: "They go to the party the following weekend." or "They are going to the party the following weekend." These are straightforward statements, with no uncertainty: They are sure to go to the party the following weekend.
Such statements can be modified, using modal verbs, to inject an element of uncertainty into the statement, equivalently: "They
The sentences I wrote are examples of reported speech. In direct speech, the first one would be: "We'll be going to a party this weekend". So will becomes would in reported speech.
My question, however, was whether the be going is an infinitive or gerund form. You mention, in your last paragraph, that it's an infinitive structure. If so, why is it followed by -ing?
You can say, equivalently: "We will go to the party this weekend." or "We will be going to the party this weekend."
In reported speech, the tense is past, so you'd say, equivalently: "He said that they would go to the party this weekend." or "He said that they would be going to the party this weekend."