When it comes to Yom Kippur, the halachah is that even chewing on something non-nutritional is prohibited. The example given is aromatic wood (Rema to Shulchan Aruch, Orach Chaim 612:6), and Mishnah Berurah (ibid. :18) explains that one does feel a sensation of flavor from doing so. So that would seem to cover your examples of a stick or sugar-free gum; I suppose you could argue that plastic is different, though.
As for grass - Rambam (Hil. Shevisas Asor 2:6) gives "bitter grasses" as an example of food not fit for human consumption, and says that someone who eats that on Yom Kippur is not subject to kares, but is punishable by makas mardus (lashes at the discretion of the beis din). Mishnah Berurah (Shaar Hatziyun 612, ד"ה אכל) explains that he means "bitter grasses" as opposed to edible herbs; I'd guess, though, that regular lawn grass would probably fall into the "bitter grasses" category (especially with all the pesticides usually applied to it...).
In connection with chewing something and spitting out the resulting saliva, Rema there references 567:3. There he says that this applies only to Yom Kippur, and that it would be permissible on other fast days. However, Mishnah Berurah (ibid. :13) says that this is correct only according to the Mordechai; according to the Mechaber (ibid. :1) this should be true of Tisha Be'Av as well, and according to the Rema (ibid.) it would be true of all public fast days.