Should one mix serif and sans-serif type fonts on the same Webpage, for plain text elements (not headlines)?
There are "rules" that sans-serif look better for headlines, serif for plain text.
Are there "rules" that specify what to do if I'd wanted to mix serif and sans-serif on plain text elements (not headlines), like Twitter does on its individual tweet pages?
Absolutely you can do this. Some of (or a lot depending on your perspective) of the principles of print definitely carry over to the web.
There have been a lot of questions about choosing type. That is a largely subjective questions, but a very valid one and I'll it the already existing questions to answer it.
UPDATE per the OP's addition:
Mixing serif and sans-serif in basal text is okay when it is used under the general rule that text set in a different type from the basal text is because it signifies something of importance to the reader. By way of example, a link within the text or a call-out to a figure somewhere on the page. It isn't enough to change the type for the sake of changing it because it will stand out from the rest of the text, like adding a spot of color to a field of grey.
Beyond that, the usual rules for selecting typefaces still applies, but this is still subjective and open to amendment. The only caveat I would think applies here is that it is typical for the basal text to be in a serif font, and any highlighted text due to special meaning would be in the sans-serif and not the other way around. The only exception to that would be in the case of math, which is invariably better displayed in serif.