Your course grade will be calculated as follows:
(1) Complete the following proof that if $m$ is a limiting parallel to $\ell$, then $\ell$ is a limiting parallel to $m$: Take a point $P_1$ on $\ell$. Let $Q$ be the point on $m$ such that $\overline{P_1Q} \perp m$. Let $P_2$ be the point on $\ell$ such that $\overline{QP_2} \perp \ell$. Use a continuity argument to show that there is a point $P$ between $P_1$ and $P_2$ such that the segment $\overline{PQ}$ makes equal angles with $\ell$ and $m$. Then consider the perpendicular bisector of $\overline{PQ}$ and use a symmetry argument. (I think that a proof along these lines is simpler than the book's proof.)
(2) Show that there is an isometry $f$ of the Poincaré model (in other words a function from the open unit disk to itself that preserves the hyperbolic distance) and a line $\ell$ in the model such that $\ell$ is (represented by) a Euclidean line segment and $f[\ell]$ is (represented by) a Euclidean arc. The point is that the property of looking "straight" or "curved" is not an intrinsic property of hyperbolic lines; it depends on how we model them in Euclidean space. Hint: our work with reflections doesn't depend on the parallel postulate, so it is still valid in hyperbolic geoemtry.
(3) Exercise 7.3.3. I don't know what the hint "use the parallelism properties of reflections" means, but anyway it should not be hard to show that if $m$ is a limiting parallel to $\ell$, then $r_\ell[m]$ is a limiting parallel to $\ell$ on the same side. The part about omega points turns out to be trivial once you untangle the definitions (including what it means for an isometry to fix an omega point) so I'll say you can skip this part, but you should think for a minute about what it means.
Remark: To visualize exercises (3) and (4) it may be helpful to consider the special case where the line $\ell$ is represented in the Poincaré model by a diameter of the unit disk. Then we can extend the reflection $r_\ell$ to act on the boundary of the unit disk in a natural way. Two and only two of the boundary points will be fixed by this extension of $r_\ell$.
(4) Exercise 7.3.4. Hint: Let $\ell'$ be a line that is not right-limiting parallel to $\ell$, which means that it has a different omega point on the right side (the same argument will work for the left side.) We want to show that this right omega point of $\ell'$ is not fixed by the reflection $r_\ell$. In other words, we want to show that the line $\ell'$ is not right-limiting parallel to its own reflection $r_\ell[\ell']$. Consider two cases: (a) $\ell'$ intersects $\ell$; (b) $\ell'$ is parallel to $\ell$ but is not right-limiting parallel to $\ell$.
(5) Exercise 7.3.10. You may use Exercise 7.3.6 even though we did not prove it (note that it's easy to prove the special case of Exercise 7.3.6 where one of the angles is a right angle, because the other angle will be an instance of the notion of "angle of parallelism", which is always less than a right angle.) Hint: you can think of the statement in Exercise 7.3.10 as an AAS congruence theorem for omega triangles, because you can think of the sides $\overline{P\Omega}$ and $\overline{P'\Omega'}$ has both having infinite length. Then think about how you can prove AAS congruence for ordinary triangles from SAS congruence for ordinary triangles.