For instance, if something non-edible such as a chemical or eraser shavings are drained down the sink, would that cause any kashrut problems?
Inedible nonbiological matter has Kashrus neutral status. It is not, nor can it become, non kosher. Completely inedible matter also has Kashrus neutral status. This would include completely inedible matter of biological origin as well as completely inedible mixtures containing biological matter (For example,detergent or drain cleaners are Kashrus neutral due to their complete inedibility, regardless of biological ingredients contained in them.)source This is because the Torah only permits or forbids the eating of food, and is never concerned with the consumption of non-food.
In rare cases, you might have an issue with blios, that is, absorption of taste from edible matter. For example, a metal pot (Kashrus neutral) used to cook non kosher food can absorb taste from the non kosher food.
the laws of kosher are discussed in the torah in parsha shmini (Lev.11).
there it discusses the kosher status of fish, animals and birds.
no mention of plants or inanimate materials. so we can assume none of those are forbidden regarding kosher