Why do heavy nuclei have half-lives if they are unstable why do they take millions of years to break down in some cases why don't they simple do it instantly? What makes them stay together?
Why do heavy nuclei have half lives?
To keep it simple -> Coulomb repulsion
Why do heavy nuclei have half-lives if they are unstable why do they take millions of years to break down in some cases why don't they simple do it instantly?
You cannot predict when one nuclei "breaks down". It can live forever. However, for a big sample you will have half of your initial nuclei after the half-life time. So it's a stochastic process. If you want a more detailed answer, you will have to deal with quantum physics. Keywords: spin, gerade, ungerade, magic numbers, tunneling, etc.
What makes them stay together?
Firstly radio-isotopes ('unstable' nuclei) show a very wide range of half-lives and each specific radio-isotope has a specific half-live. The range of half-lives varies from extremely small (milliseconds and smaller) to extremely large (millions of years in some cases). Stable, non-radioactive isotopes can be considered for simplicity sake to have half-lives of near $\infty$ (infinite) value.
Radioactive decay is essentially a quantum-nuclear phenomenon. Look for instance at this interesting approach at modelling $\alpha$ decay of Polonium-212 (half-life 0.3 microseconds). Quantum tunnelling causes the $\alpha$ particle to stray out of the nucleus with a certain probability, calculated on that web page. The probability of finding the $\alpha$ particle outside of the nucleus is directly related to the half-live of Po-212, as explained on the same site: the higher the probability the smaller the half-life.
In the case of radio-isotopes that undergo $\alpha$ decay with much longer half-lives, the probability of finding the particle outside of the nucleus is much smaller.