The given chemical equation represents a first-order reaction:

$$2N_2O_5 (g) \to 4NO_2 (g) + O_2 (g)$$

In first-order reactions, the rate of reaction is directly proportional to the concentration of the reactant. The rate law can be expressed as:

$$Rate = k[N_2O_5]$$

where k is the rate constant.

The integrated rate equation for a first-order reaction is:

$$\ln [N_2O_5] = -kt + \ln [N_2O_5]_0$$

where [N_2O_5]_0 is the initial concentration and [N_2O_5] is the concentration at time t.

Rearranging gives:

$$[N_2O_5] = [N_2O_5]_0 e^{-kt}$$

Now, let's analyze each option:

Option A: "the concentration of the reactant decreases exponentially with time".

This matches the expression $$[N_2O_5] = [N_2O_5]_0 e^{-kt}$$ which shows an exponential decline in the concentration of the reactant. Therefore, option A is correct.

Option B: "the half-life of the reaction decreases with increasing temperature".

In first-order reactions, the half-life is determined by the equation $$t_{1/2} = \frac{0.693}{k}.$$ Increases in temperature typically increase the rate constant k, thus decreasing the half-life. However, it is not directly related to the half-life itself but rather to the rate constant via the Arrhenius equation. Nonetheless, the essence of the statement is correct as increasing temperature generally decreases the half-life.

Option C: "the half-life of the reaction depends on the initial concentration of the reactant".

The half-life for a first-order reaction given by $$t_{1/2} = \frac{0.693}{k}$$ is independent of the initial concentration [N_2O_5]_0. Therefore, option C is incorrect.

Option D: "the reaction proceeds to 99.6% completion in eight half-life durations".

Completion to a certain level can be found using the formula for decay over multiple half-lives, given as $$[N_2O_5] = [N_2O_5]_0 \times \left(\frac{1}{2}\right)^n$$ where n is the number of half-lives. For eight half-lives, we have:

$$Final\, Concentration = [N_2O_5]_0 \times \left(\frac{1}{2}\right)^8 = [N_2O_5]_0 \times \frac{1}{256}$$

The reaction proceeds to $$\frac{255}{256} \approx 99.6\%$$ completion. Thus, option D is correct.

In summary, options A, B, and D are correct, while option C is incorrect.