$\def\F{\mathscr{F}}\def\Ω{{\mit Ω}}\def\B{\mathscr{B}}\def\R{\mathbb{R}}$For Q1, since a stopping time are intuitively the (random) time when one stops observing the stochastic process according to some given rules with information available up to that time, then $\F_τ$ is the collection of events that can be checked to see whether they have happened or not by observation up to $τ$. Now for this remark:
$A\in \F_τ$ means that even if $A \notin \F$, you know that $A$ occur or not whenever $τ \leqslant t$.
The wording is incorrect because the definition of $\F_τ$ requires that $A \in \F$ for any $A \in \F_τ$, but what your professor said is more likely to be this:
$A\in \F_τ$ means that even if $A \notin \F_{\color{red}{t}}$, you know that $A$ occur or not whenever $τ \leqslant t$.
This interpretation is indeed correct since $A \cap \{τ \leqslant t\} \in \F_t$ exactly means that the knowledge on whether the event $A \cap \{τ \leqslant t\}$ has happened or not is available in the information up to time $t$.
For Q2, progressive measurability of $X$ is usually assumed to prove that $X_τ$ is $\F_τ$-measurable. Assuming this, the mapping$$
\begin{matrix}
([0, t] × \Ω, \B([0, t]) × \F) & \longrightarrow & (\R, \B(\R))\\
(s, ω) & \longmapsto & X(s, ω)
\end{matrix}
$$
is measurable for any $t \geqslant 0$. Note that$$
\begin{matrix}
(\Ω, \F) & \longrightarrow & ([0, t] × \Ω, \B([0, t]) × \F)\\
ω & \longmapsto & (τ(ω) ∧ t, ω)
\end{matrix}
$$
is also measurable since $τ ∧ t$ is also a stopping time, then the composition$$
\begin{matrix}
(\Ω, \F) & \longrightarrow & ([0, t] × \Ω, \B([0, t]) × \F) & \longrightarrow & (\R, \B(\R))\\
ω & \longmapsto & (τ(ω) ∧ t, ω) & \longmapsto & X(τ(ω) ∧ t, ω)
\end{matrix}
$$
is measurable, which implies that the stopped process $\{X_{τ ∧ t}\}$ is progressively measurable.
Now for any $t \geqslant 0$ and $B \in \B(\R)$, because $\{X_{τ ∧ t} \in B\} \in \F_t$, so$$
\{X_τ \in B\} \cap \{τ \leqslant t\} = \{X_{τ ∧ t} \in B\} \cap \{τ \leqslant t\} \in \F_t.
$$
Thus $X_τ$ is $\F_τ$-measurable.
Best Answer
I think that I can give a motivating example that shows what kinds of problems arise for random times that are not stopping times, i.e. $\tau:\Omega\to[0,\infty)$ such that there exists $s$ when $\{\tau\leq s\}\notin\mathcal{F}_s$. Maybe you've seen this example, so I apologize if so.
So, here's the example. Say that I am gambling, and that $M_t$ is my amount of money at time $t$. It would be nice if I could know when I should quit, or in other words the time when I have the most money: \[\tau=\inf\ \{ t : M_t\geq M_s \forall s \}.\]
I claim that $\tau$ is not a stopping time. To see this, note that the event $\{\tau\leq s\}$ is the same as the event $\{\exists t\leq s: M_t\geq M_q\forall q>s\}$. This depends on $M_q$ for $q>s$ so this event is clearly not contained in $\mathcal{F}_s$. The stopped process $M_\tau$ represents my maximum cash if I keep playing, and I would really like to know it, but it's not something I could ever actually expect to know.
(This also addresses something in your question -- in this example, the event $\{M_t=M_\tau\}$ is definitely not contained in $\mathcal{F}_t$. For this example that is the same as the event $\{\tau=t\}$, which is the event that I should stop playing right now.)
So, to me this gives an intuition for why people care about stopping times. The idea behind the definition "$\tau$ is a stopping time $\iff$ \[ \{ \tau=t \} \in\mathcal{F}_t\] for all $t$" is that $\tau$ should not make use of information from the future. So, it helps us model quantities that correspond to measurable events in our filtration. If the filtration is meaningful as part of a model of the world, then that's helpful.
The other thing is that the definition of stopping times allows for optional stopping theorems.
I think that maybe looking at an application could be helpful too, since all of these gambling examples can get a little boring. If you've ever seen Erdos-Renyi random graphs, then maybe this paper studying the phase transition to the existence of a giant connected component using martingales and stopping times could be of interest. This paper makes use of optional stopping type theorems.