Chemical elements can be unstable. They are not the "true" atoms.
Conservation of mass is not valid. The quantity that is really conserved is the total energy.

What are the implications of the existence of unstable elements on the age of the earth, the age of the Universe? How are they made in nature?

Nuclear energy is sufficiently large to account for the energy output of the sun.
 
Can the energy within the nucleus be extracted in a manner analogously to the electromagnetic energy stored in the electronic structure of atoms?

Nuclei are made of protons and neutrons. Are these true atoms? There was good reason to doubt they were, even in the 1930s.

There are strong forces, newly discovered operating in the atomic nucleus, which are responsible for holding the nucleus together despite the tremendous electrical repulsion among the protons.

There is another new force, called the weak force, responsible for beta radioactivity. This is a bizarre force because it seems to violate the conservation of energy and momentum, unless one includes an elusive particle called the neutrino.
 
 

Electromagnetic force	Gravitational force
Strong Nuclear force	Weak Nuclear force

Is the neutrino real? Is it just a "deus ex machina" to save the conservation laws? Fermi's theory fits the experiments well, but why don't we have a direct confirmation of the neutrino?

Anti-particles were predicted by relativistic quantum mechanics. The positron was discovered by Carl Anderson in 1932. How many other particles are waiting to be discovered? Is there really such a thing as an atom?