Does the Independence Axiom Require Statistical Independence? First: Given this definition of the Independence Axiom,

If for all $P$, $P'$, $P''$ in the set of lotteries over outcome space $X$, when:
$P$ preferred to $P'$ $\implies$ $aP + (1-a)P''$ preferred to $aP' + (1-a)P''$ for all $a$ in $(0,1)$.&quot;

Can I do what I do here?
Then $P$ preferred to $P'$ $\implies$ $P$ preferred to $aP' + (1-a)P$ for all $a$ in $(0,1)$.
Second:
Does satisfaction of the Independence Axiom rely on the statistical independence of the lotteries involved? It doesn't seem to be mentioned anywhere I've looked.
Any direction that can be offered is greatly appreciated.
First Question:

Yes, you can obtain $P \succsim P' \Rightarrow P \succsim a P' + (1-a)P \forall a\in (0,1)$ from setting $P''=P$ in the axiom. Thus, your new condition is a special case of the independence axiom as stated.

Second Question:

Statistical independence of $P,P',P''$ is not assumed in the independence axiom. The notion of statistical independence of lotteries is not even meaningful since you can only choose one lottery. Only when you would choose multiple lotteries at the same time their dependence would matter since you may be able to reduce/increase risk by combining lotteries which are not independent.