What is the half-life threshold for an isotope to be considered stable? What minimum half-life an isotope should have to be considered stable?
 A: It depends on the context.
For a given number of nucleons, there are typically one or two charge states that have much lower energy than any of their neighbors.  For example, in the $A=12$ isobar, the weak decays $^{12}\text{B}\to{}^{12}\text{C}$ and $^{12}\text N\to{}^{12}\text C$ both release around 15 MeV excess binding energy.  Carbon-12 is at the bottom of that valley.  Carbon-12 also is more tightly bound than any $A=11$ system with a free proton or neutron at rest.  So carbon-12 is really and truly stable: unless there is proton decay (for which there's no evidence in the free proton, and which we'd expect to be even slower for a bound proton) there's no known final state to which it could release energy by decaying spontaneously.  This is the case for most of the isotopes with black squares in the table of nuclides.
Isotopes with lifetimes comparable to or longer than the age of the earth occur naturally and can be considered stable for most practical purposes: for instance, uranium-containing minerals can be smelted into uranium metal and machined.
In fast reactions, an long-lived unstable nuclide may effectively be considered stable.  This is, for instance, why cosmic rays at sea level consist of muons and not pions.  It's also why the entire ensemble of neutron-rich isotopes is necessary to understand the rapid fusion process in supernovae.
There do exist nuclei for which decays are allowed energetically but have not been observed.  Brandon Enright mentions bismuth 209.  Another example is xenon 124, which is too tightly bound to decay to iodine 124 and a positron-neutrino pair at rest, but can energetically decay to tellurium 124 and two positrons at once.  This is a system that is used to search for evidence that the neutrino is its own antiparticle.
