What are the halachic and factual issues involved in, and various rabbis' holdings on, whether flavored mouthwash, flavored toothpaste, and flavored lip balm need to be kosher, or need a hechsher (kosher certification)? For example, one argument alluded to in an answer is "it's inedible so it's fine": what are the halachic parameters of "inedible", sources to say that such inedibility renders the product okay (or not) for use, and factual evidence that the product is indeed inedible (or not)? And likewise for other arguments/issues.
Toothpaste -- If I recall correctly, Adwe had asked Rabbi Schwob (or was it Breuer?) for certification -- he refused to certify it. He explained that toothpaste is inedible and thus does not need certification, and if he certified one brand which might later go out of business, people would think non-hechshered toothpaste was prohibited.
I've recently seen some additional discussion about toothpaste -- if you say "it's inedible so it's fine", you're all set. But for those who wanted to be stricter, a generation ago (I think this is attributed to R' Yaakov Kaminetsky?) it helped that toothpaste by volume was >51% water and calcium carbonate, ingredients that posed no kosher issues. Glycerin and the like were minority by volume. Today the bulking agent has switched from calcium carbonate to hydrated silica, of which a little goes a long ways, so we're no longer guaranteed > 51% non-problematic ingredients. (Not that hydrated silica is a kosher concern -- it's just that there's less of it to out-volume the problematic ingredients.)
From this article in The Kosher Spirit (OK's Kashrus Magazine):
Toothpaste does not require a kosher symbol, but it is considered a chumra to buy toothpaste with a kosher symbol. [...] [lipstick]/Chapstick do not require a kosher symbol.
I also saw that according to the Chicago Rabbinical Council, all Lip Balms are recommended, but preferably without flavor.
For a little bit on the "why" here are some excerpts from this article from Star-K:
We are familiar with the Torah law prohibiting one from eating non-kosher food. This halacha only prohibits the consumption of non-kosher food. One may, however, derive benefit from non-kosher food.
And why we aren't worried about swallowing trace amounts of lipstick:
This is true even though one may inadvertently swallow traces of lipstick that mix with food. These trace amounts of lipstick are batel b'shishim (1:60 or less). [from footnote:] It is also a davar she'aino mischaven with no psik raisha
And about toothpaste:
If there is no intention to swallow a non-food product (e.g. toothpaste, mouthwash, floss), one need not worry about the ingredients [from footnote: it is not fit for human consumption and there is no Achshivay]. Nonetheless, it is a hiddur to use kosher products if they might inadvertently be swallowed.
HaRav Yitzhak Abadi Shelit"a writes (Sefer Or Yitzhak I YD 24) that toothpaste doesn't need to be kosher.