It is usually desirable to make titles and quotations stand out from your own text. So saying The Sunday Times article The Price of Fish achieves this objective quite well.
But it is not obligatory - it is a matter of choice.
Yes, when referring to the title of a magazine article, it is common practice to put it in quotation marks. This helps to distinguish the title from the rest of the text.
No.
You use quotations for short stories, poems, article, and songs I believe. Everything else is underlined.
Quotations for article titles. Italics and underlining are for full books - and the titles of journals.
When referencing a journal article in a paper, you typically do not use quotations. Instead, you should use in-text citations and provide a full reference in the bibliography or works cited page. The citation style may vary depending on the formatting requirements of the paper or publication.
If you mean newspaper or magazine articles, and if you are using MLA format (i.e., for an English class), then article titles should be in quotation marks rather than underlined or italicized. APA or Chicago formats may have different requirements; I'm not familiar with them.
An online magazine article is the same as an article in a paper magazine except you can find it online, usually through the magazine's own website or through an article aggregation site. You can find an example through the given link.
Quotations should fit into sentences:contextually.grammatically.
Smithsonian
Yes; the article title should be placed inside quotation marks, while the name of the newspaper or magazine is italicized.
the magazine Living Without
Magazines are typically italicized rather than underlined or placed in quotation marks. For example, "National Geographic" or Time.