$\def\F{\mathscr{F}}\def\Ω{{\mit Ω}}\def\B{\mathscr{B}}\def\R{\mathbb{R}}$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
$A \in \mathcal{F}_t$ means: Given the information up to time $t$, we can decide for some fixed $\omega \in \Omega$ whether $\omega \in A$ (or $\omega \in A^c$), i.e. whether the event $A$ happens (or not).
The interpretation for $A \in \mathcal{F}_{\tau}$ is very similar: Given the information up to time $t$ and given that the stopping time $\tau(\omega)$ already occured before time $t$, we want to decide whether $\omega \in A$ (or $\omega \in A^c$).
Example: Let $(B_t)_{t \geq 0}$ be a stochastic process with continuous sample paths and $B_0=0$. Define
$$\tau := \inf\{t>0; B_t \notin (-1,1)\}.$$