Xenon is so expensive it's reasonable to think about recovering it from spent nuclear reactor fuel (it's a major fission product with no long lived isotopes, although you have to very rigorously separate it from krypton, which does have some.)
Xenon isn't inert; it can form chemical compounds like xenon difluoride. In the body it's probably dissolving into cell membranes or other fatty components.
What do you mean by "no long-lived isotopes"? That fission only produces short lived Xenon isotopes? Then, what use would it be to harvest it from fuel?
For applications like in the article you want very long-lived or ideally stable Xe isotopes, which do exist.
The short-lived stuff decays away, leaving the safe stuff. When the isotopes decay away, they become other elements that are easily chemically separated. Or so I'd guess.
That's right. The xenon isotopes that are left are stable.
Another fission product that might be worth recovering is rhodium, although not being a gas the process would be more involved.