This is a question from my lecture notes and I'd be grateful if you could tell me whether my solution to $(iii)$ is correct in particular:
If $\tau=t$ that means that $\forall\omega\in\Omega, \tau(\omega)=t$. Then $\{\tau\leq t\}=\Omega $ hence $\mathscr{F}_\tau=\{A \in \mathscr{F}: A \bigcap\Omega \in \mathscr{F}_t\}=\{A \in \mathscr{F}: A \in \mathscr{F}_t\}=\mathscr{F}_t$
However, now the event $\{\tau< t\}=\emptyset $ and I can't really define $\mathscr{F}_{\tau+}$
I think you've misinterpreted the definitions and are treating $t$ in the definitions as fixed. We have $\mathcal{F}_\tau = \{A \in \mathcal{F}: A \cap \{\tau \leq s\} \in \mathcal{F}_s \mbox{ for every } s \geq 0\}$ and the other definitions change in a similar way.
So for example if $\tau = t$ a.s. then $\mathcal{F}_\tau = \{A \in \mathcal{F}: A \cap \{t \leq s\} \in \mathcal{F}_s \mbox{ for every } s \geq 0\}$. So if $A \in \mathcal{F}_\tau$ then $A \cap \{\tau \leq t\} = A \in \mathcal{F}_t$ and conversely if $A \in \mathcal{F}_t$ then $A \cap \{t \leq s\} = \emptyset \in \mathcal{F}_s$ for $s < t$ and $A \cap \{t \leq s\} = A \in \mathcal{F}_t \subset \mathcal{F}_s$ otherwise so $A \in \mathcal{F}_\tau$. Hence $\mathcal{F}_t = \mathcal{F}_\tau$.
Having correctly interpreted the definitions showing that $\mathcal{F}_{t+} = \mathcal{F}_{\tau +}$ when $\tau = t$ a.s. is a similar exercise and so I leave it to you.