Questions about the Supremum and the Empty Set

In Rudin's Principles of Mathematical Analysis there are these two definitions:

Definition 1.8:Suppose $S$ is an ordered set, $E \subset S$, and $E$ is bounded above. Suppose there exists an $\alpha \in S$ with the following properties:

1. $\alpha$ is an upper bound of $E$
2. If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $E$.

Then $\alpha$ is called the least upper bound of $E$ or the supremum of $E$, and we write $\alpha = \sup E$.

Definition 1.10: An ordered set $S$ is said to have the least-upper-bound property if the following is true:

If $E \subset S$, $E$ is not empty, and $E$ is bounded above, then $\sup E$ exists in S.

My Questions

I am having a hard time thinking about the technical details regarding the empty set and the supremum. Consider the following situation: $S = \mathbb{R}$ and $E = \emptyset$. I understand that $\forall x \in \mathbb{R}, \forall y \in \emptyset, y \leq x$ is true. Therefore every real number is an upper bound for $\emptyset$. I don't think the empty set can have a supremum because if we suppose it has one then we can find a smaller upper bound which would give a contradiction. Is this correct?

If $S = \emptyset$ and $E = \emptyset$ what would happen in this situation? Would the empty set have a supremum? Couldn't we form a statement that said every element in the empty set is the supremum of the empty set, which would be a vacuous truth?

What if $S = [0, 1]$ and $ E = \emptyset$. In this situation, $S$ has a minimum so could we find a least upper boumd even though $E$ is empty?

I'm having a hard time because of the vacous truths and the definitions saying that the supremum "exists". If you could provide an explanation for the situations I described above this would help me understand the supremum better. The situation where S and E are the empty set is the most confusing. Thank you for your help!