The second question of the OP is an important one, because it asks for clarification on a matter that I have never seen explicitly clarified.
The Independence Axiom is defined over simple lotteries. A simple lottery is a set of probabilities that add up to one, and the fixed values ("outcomes") associated with each probability. But these together form a (marginal) statistical distribution. So we can think of a simple lottery as a random variable characterized by/following this distribution.
Let $L'$, $L''$ and $L'''$ be three such random variables. The preferences of a decision maker that obeys the Independence Axiom satisfy
$$L \succsim L' \;\;\text {if and only if}\;\; aL+(1-a)L'' \succsim aL' + (1-a)L'',\;\; a\in (0,1)$$
Note that the Independence Axiom talks about preferences over random variables. $L \succsim L'$ is translated "I prefer random variable $L$ to random variable $L'$", which in more detail means "I prefer that my fortunes are submitted to the uncertainty represented by random variable $L$ than to the uncertainty represented by random variable $L'$." We do not ask why the decision maker prefers that. We do not ask what is the statistical relationship between the two, we just take the preference statement as a primitive given. Then, we say that under the Independence Axiom, "if this is so, then the RHS preference relation also holds, and vice versa".
How are we to translate verbally the expression $aL+(1-a)L''$, or $aL' + (1-a)L''$? The crucial issue here is the interpretation of $a$, as well as to what mathematical operation the expression $aL$ translates into.
What is $a$? Is it a fixed, deterministic "mixture coefficient"? Or is it a probability, and so $aL+(1-a)L''$ is a compound gamble?
The distinction is, again, crucial: if $a$ is a constant and $aL$ represents a multiplication of values with the values (outcomes) $L$ takes, then $aL+(1-a)L''$ is plain convex combination of two random variables, and analogously for $aL' + (1-a)L''$. But then, we cannot rationally defend the Independence Axiom, exactly because there may be stochastic dependence involved between the three simple lotteries such that it may reverse the preference between the two convex combinations. Where moreover $aL$ would mean "multiplication of the random variable $L$ by $a$". But this is not the case (See this post as regards the interpretation of $aL$).
The "optimal portfolio choice" (which from a comment appears to be the concern of the OP), falls into such a framework, and we can see that it cannot be linked to the Independence Axiom.
Continuing, we are necessarily led to view $a$ as something else than a constant. In fact it is treated as a probability, and so the mixture as being a compound gamble. Is a compound gamble also a random variable? It is, but of different nature: if $a$ is meant to reflect a probability, a new source of randomness appears. $a$ here is the "probability of success" of a Bernoulli rv, say $B$, which moreover, is statistically independent from the three simple lotteries. But then ex ante we are essentially looking at the random variable
$$B\cdot L + (1-B)\cdot L'',\;\; B \sim Bern(p=a)$$
and likewise for the other combination (and this is the correct way to write it, in the notation of mathematical statistics). Here the Independence Axiom acquires some intuitive justification:
"Assume that you prefer $L$ to $L'$. Then, if you were asked "what do
you prefer: to be subjected to "either $L$ or $L''$" with some probability
structure (the $B$), or to be subjected to "either $L'$ or $L''$" with the
same probability?" It appears reasonable to say "since I prefer $L$ to
$L'$, then I prefer the situation where $L$ is present, rather than
the one where $L'$ is present".
Unfortunately, this discussion is not explicitly made in the microeconomic books I know of, and moreover the standard notation is with the $a$'s rather than with the Bernoulli rv.
Sometimes there is some discussion on the matter. Consider for example Mas-Colell, Whinston-Green, p. 172. After stating the Independence Axiom in p. 171, they write (bold my emphasis)
"Assume for example that $L \succsim L'$ and $a=1/2$. Then $aL + (1-a)L'"$ can be thought of as the compound lottery arising from a
coin toss in which the decision maker gets $L$ if heads comes up and
$L''$ if tails does."
But this description essentially says that $a$ here is the random variable $B$, not some fixed deterministic mixture coefficient.
And a few lines below
"In the theory of consumer demand, for example, there is no reason to believe that a consumer's preferences over various bundles of goods 1
and 2 should be independent of the quantities of the other goods that
he will consume.(...) In the present context however (...) this other outcome, $L''$ should be irrelevant
to his choice because, in contrast with the consumer context, he does
not consume $L$ or $L'$ together with $L''$ but rather, only instead
of it (if $L$ or $L'$ is the realized outcome)"
This essentially says again that $a$ in reality is not a mixture constant, but it represents a binary random variable independent of the simple lotteries (our $B$). Unfortunately, the notation used (the $a$) is a notation usually reserved for constants. Moreover the expression $aL$ does not imply multiplication of $L$ by some constant $a$, but it is equivalent to the product random variable $B\cdot L$, with $B$ a Bernoulli random variable and $P(B=1) = a$.
