Preference over lotteries without independence axiom
No, not necessarily. Without the independence axiom (or something else to replace it) there is not much you can infer about preferences over (non-degenerate) lotteries from knowing preferences over outcomes only.

For instance, let $p^L_n$ be the probability of outcomes $n \in \{1, 2, 3\}$. Then preferences  over lotteries $\succeq^*$ represented by the utility function

$$U(L) = p^L_1 + \beta [p^L_2p^L_3],$$

are continuous  and rational, but do not satisfy the independence axiom. For $\beta$ large enough, it is not even the case that $(1,0,0)$ is the best lottery, although $(1,0,0) \succ^* (0,1,0)$ and $(1,0,0) \succ^* (0,0,1)$.

To see why, observe that

$$ U(1,0,0) = 1, $$
$$ U(0,1,0) = 0, $$
$$ U(0,0,1) = 0, $$

However, for $\beta &gt; 4$,

$$ U\left(0,\frac{1}{2},\frac{1}{2}\right) &gt; 1 .$$

Violation of the independence axiom can be seen from the fact that, when $\beta &gt; 4$,

$$ [1,0,0] \succ [0,1,0]  ,$$

although

$$ \left[0,\frac{1}{2},\frac{1}{2}\right] \succ \left[ \frac{1}{2}, 0, \frac{1}{2}\right]. $$