Can we given an example of an element $m \in M$, where $M \neq \emptyset$ is a partially ordered set, such that $m$ is a maximal element of $M$, but $m$ is not an upper bound of any subset $W (\neq \emptyset)$ of $M$?
Can we have an example of a non-empty subset $W$ of $M$ with more than one upper bounds?
I know that there may not be any maximal element in $M$: as an example, let $X$ be a given infinite set, and let $\mathscr{P}(X)$ denote the power set of $X$ (i.e. the set consisting of all the subsets of $X$); let $M$ be the set $\mathscr{P}(X) - \{ X \}$. Then $M$ has no maximal element. Am I right?
I also know that not every maximal element is an upper bound: let $X = \{1, 2 \}$, and let $M = \{ \emptyset, \{1\}, \{2 \} \}$ ordered by set inclusion; then $M$ is partially ordered. Both $\{1\}$ and $\{2 \}$ are maximal elements of $M$, but the subset $M$ of $M$ has no upper bound in $M$. Am I right?
Let $m \in M$ be maximal element. Consider $W := \{ n \in M :~~ n \leq_M m\}$ then $m \in W$ so $W$ is non-empty. But $m$ is an upper bound for $W$ since it is comparable with any $w \in W$ and greater than all of them.
About your second question : consider $M := \mathbb{R}$ with the usual order and $W := [0,1]$. Then any real number greater or equal to 1 is an upper bound for $W$ i.e. any $r \in [1, \infty)$ is comparable with any $w \in W$ and $w \leq r$.
Your last two comments are OK. You are right.