We assume that $X \sim \text{Unif}\,(a, 2a)$ on $[a;2a]$ where $a>0$ and $\{ x_i\}_{i=1}^{N}$ are i.i.d realizations from this distribution.
How can I prove that the estimator $\hat{a} = \text{min}(x_i)$ is well-defined? Meaning that for all $i$, $$\hat{a} \leq x_i \leq 2\hat{a}$$ holds?
Can it be also proved that the estimator $\hat{a} = \frac{\text{max}(x_i)}{2}$ is also well-defined such that: $$\hat{a} \leq x_i \leq 2\hat{a}$$ holds as well for all $i$?
First assume that $\hat a = \min \{ x_1, \dots, x_n \}$. Since $a \leq \hat a \leq x_i$ for $i = 1, \dots, n$, it follows that $2a \leq 2 \hat a$, and since $x_i \leq 2a$ for $i = 1, \dots, n$, we see that $\hat a \leq x_i \leq 2\hat a$ for $i = 1, \dots, n$.
Your second part about $\max$ makes no sense. How could every $x_i$ be at least as big as the maximum? I think what you are reaching for is $\hat a / 2 \leq x_i \leq \hat a$, assuming that $\hat a = \max \{ x_1, \dots, x_n \}$.
In the future I suggest using different notation for $\min$ and $\max$.
If you are trying to find the MLE, check out question 3 on page 4 here: STAT 135 Solutions to Homework 4.