First year uni maths - geometry

For #1 the fact that they're in the hyperbolic plane is irrelevant; just find the points of intersection as if you're just working in the upper half-plane with the Euclidean metric.

For #2, set $z=\lambda(z)$ and solve for $z$. You might get more than one possible answer, but one of them won't lie in H.

I'm unsure as to how to find the intersections

It's probably easier to work with Cartesian coordinates. For example, to find where a and b intersect note that $|z-1|=3$ is a circle with centre 1+0i and radius 3, so we can write this as $(x-1)^2+y^2=9$. The line $\text{Re}(z)=3$ is precisely the line $x=3$, and so these intersect whenever $(3-1)^2+y^2=9$, which you can solve for $y$. You get a positive and negative solution for y, so it should be clear which lies in the upper half-plane and which doesn't. You do a similar thing for the other ones.

The question says that I have to find the complex numbers which represent those vertices. With the method quoted, I end up with plus or minus root 5. But that doesn't give me a complex number