C

Step 1: Analyze Statement I
All radioactive elements indeed decay exponentially with time due to the nature of radioactive decay. The decay of a radioactive substance can be described by the equation:
$$N(t) = N_0 e^{-rac{t}{T_{1/2}}}$$
where $N(t)$ is the quantity of the substance left at time $t$, $N_0$ is the initial quantity, $T_{1/2}$ is the half-life, and $e$ is the base of the natural logarithm. Thus, Statement I is true.

Step 2: Analyze Statement II
The half-life of a radioactive element indeed refers to the time required for half of the radioactive atoms to disintegrate. This definition is universally accepted and correct. Hence, Statement II is true.

Step 3: Analyze Statement III
Radioactive dating, such as Carbon-14 dating, is a method used to determine the age of objects, including the earth, based on the ratio of parent and daughter isotopes. Therefore, Statement III is also true.

Step 4: Analyze Statement IV
The half-life of a radioactive element is not 50% of its average life period. In fact, the average life (or mean life) is given by the relation:
$$T_{avg} = T_{1/2} imes rac{2}{ ext{ln}(2)}$$
which shows that $T_{avg}$ is approximately 1.442 times the half-life. Thus, Statement IV is false.

Conclusion
Since Statements I, II, and III are true, while Statement IV is false, the correct answer is option C: I, II and III.