Old question, but I came in search of an answer and have some things to share on this topic!
Adobe InDesign uses character or paragraph styles to format italics and bold, using a relevant font family, this is correct. There used to be a way to visually "force" a style, but I'm using Creative Cloud and have the most recent InDesign, which doesn't show these options anymore.
What would happen if I snubbed the new rules and published an epub
with bold and italics tags? Would it create some technical problems
that would come back to haunt me? Will Amazon or other publishers
reject epubs that have traditional bold/italic tags?
It wouldn't be a "bad" thing to use HTML tags to indicate bold and italic. Ebooks can be formatted using HTML, and these tags are widely supported, even in books that are done in HTML and then converted to the relevant file formats. The text styling is transferred.
I wonder if there's a way to instruct an epub to disallow fonts that
don't "recognize" bold/italics tags. Fonts like Arial and Times New
Roman would be allowed, while really funky fonts would be essentially
banned (except perhaps in headings or art work).
The font families that are used on e-readers generally include the capability to have bold, italic, and underline styles put on them. A publisher can define a font face for an ebook, which the user may decide to use. But the joy of e-readers is that the user can change it to suit their reading preferences, such as size and font face. There is really not much point in defining what font faces are "permitted" in an epub, because the epub is not the file in control of the fonts on the device.
Perhaps the rules have changed again, but Amazon KDP supports
<i> (and many more HTML tags).
Finally, I wouldn't advise creating an ebook in InDesign. Ebooks are supposed to be "reflowable" and InDesign will place line breaks into text in order to fit it in the relevant frames. Since ebooks are supposed to work with any width of the body of the text, this is counter to InDesign's functions in size-dependent layout design. Calibre is useful for converting, but Microsoft Word and simple HTML are also options for creating an EPUB.