Pope's The Rape of the Lock is a satire primarily on the English upper class of his time. The superficiality of the characters and their obsession (both that of the men and women) with appearances are ridiculed with language more appropriate to an epic poem—and in many cases drawn from Homer and Virgil—about serious matters such as bravery and war. Even Queen Anne is mildly spoofed in this couplet:
Here thou, great Anna, whom three realms obey,
Dost sometimes counsel take--and sometimes tea.
John Gay's The Beggar's Opera satirizes the ruling class as well, but in a quite different way, by showing the criminal underworld of the time and implying those who live in it are no worse in their behavior than the upper classes are. Gay also ridicules what he considers the pretensions of the aristocracy in its obsession with Italian opera. The operas of Handel and other composers, sung in Italian before English-speaking audiences (as is the case still today, although now with the use of subtitles!), were a fashionable entertainment in London. Gay and other English literary men and intellectuals, many of whom had little feel for music, regarded opera as a somewhat ridiculous art form. The Beggar's Opera, sung in English, with low-life characters and a deliberately silly and contrived plot (with an artificially tacked-on "happy ending" as most actual operas of the time were "required" to have) is both a spoof of the operatic genre and a commentary on English society. In the twentieth century, it was recreated in the Brecht-Weill collaboration The Three-penny Opera, which includes the still well-known song "Mack the Knife."
Regarding your last point, I would agree that there is a degree of misogyny in Pope's portrayal of Belinda. She seems rather shallow and superficial, though so do the men depicted in The Rape of the Lock. However, Pope regards his own poem as "redeeming" Belinda in some sense, for in his closing lines he states that
This Lock, the Muse shall consecrate to Fame,
And midst the stars inscribe Belinda's Name.