No -- all minerals come from food and drink.
I'm defining mineral == a small molecule that has a required (generally)metallic atom as part of it.
The whole subject is VERY complex. The story for zinc covers most facets you asked about:
For just zinc see this article:
https://ods.od.nih.gov/factsheets/Zinc-HealthProfessional/
Some minerals have storage like calcium -> bone tissue. Zinc has no special storage mechanism.
Minerals can act as anti-nutrients for other minerals, preventing uptake. Zinc and magnesium play games with each other, based on intake amounts.
Massive doses of a mineral can be toxic - mostly because it prevents uptake of another mineral element. And sometimes can interfere with metabolic pathways that normally would not have been affected by lesser amount of the nutrient mineral.
Minerals like iodine have a VERY restricted required range for intake. Too high=toxic, too low == mental retardation in children. Large areas of the world have very low iodine levels in soils, so low levels in foods. India is an example.
Some minerals like selenium have a lot less known about them than other minerals.
There are also some "wannabe" minerals that are promoted by pill sellers, which do not have scientific support - vanadium is one. And vanadium studies are sort of rare anyway, IMO.