The only help I'll give here is for #2, for which I'm using Griffiths's excellent QM book (Chapter 2) as a reference. If you can get your hands on that book, buy it!
When you analyze the finite square well, after solving the Schrodinger equation, imposing boundary continuity conditions, and cleaning up the notation using a dummy variables "Z" and "Z0", you get an equation for Z that has tan(Z) on one side and sqrt{(Z0/Z)^2-1} on the other side. This can't be solved traditionally; the best you can do is plot both functions together and see where they intersect. That's why it's called a transcendental.
It looks like your teacher is giving you the hydrogen atom, since the mass of the electron is given, along with the traditional 1 angstrom distance from the nucleus.