Note: 1 question per q&a. This response addresses the question in the title.
It is somewhat difficult to gauge precisely what is meant by the “popularity” of a play and how one might measure it. There is an entire discipline of literary studies referred to as “reception studies” devoted to how one studies the history of how a work has been read and studied through its entire history. Since Google is the most widely used search engine, a survey of Google results is a starting point for statistical analysis of modern reception.
A Google search on the term “Othello, Shakespeare” 11,900,000 hits. This suggests that it is a relatively common search term. By contrast, “Macbeth, Shakespeare” returns 16,800,000 results, suggesting that Macbeth may be somewhat better known than Othello. A less well-known play such as Coriolanus returns 1,610,000 results and Titus Andronicus, 2,960,000. These results mirror what would be found in a search of Google Scholar, which returns 37,300 for Othello, 52,500 for Macbeth, and 13,000 for Coriolanus. In contrast, a non-Shakespearian work, such as Webster’s “Duchess of Malfi” returns only 2,900 results on Google Scholar and 784,000 on Google.
These statistics suggest that Othello is in the mid-range of popularity for Shakespeare’s plays, but that Shakespeare’s plays are significantly more widely known in the 21st century than other plays of the Elizabethan era.
One starting point for investigation of the reception of Othello is Roland Barthes’ statement “literature is what gets taught”, i.e. that a significant predictor of the popularity of a play is the degree to which it is a curricular standard.