1. So what is the Hamiltonian for a particle in a box? All particles in a box have the same form for the Hamiltonian, in terms of the mass and width of the potential well. Then use the mass of the electron and your width. Basically this is just plug'n'chug.
2. Start by solving the time-independent S-equation: . Once you have that, then use the time evolution operator to find solutions for the time-dependent S-equation. (We can get away with this because the potential is not a function of time.)
3. The energies will depend on a integer, usually chosen to be n. So n = 1, 2, 3, 4, 5 in your energy eigenvalue equation. Then if the electron is in n = 4, it can fall to 3, 2, and 1. So find the difference in energy between the levels and then .
4. Yeah, golf balls don't do well in QM, but you have to prove you can't see the energy quantization. Think of it this way: What is the approximate mass of the golf ball? The golf ball has to sit in a potential well that is at least as wide as the golf ball. So what is the diameter of a golf ball? (Order of magnitude numbers should work just fine here. You don't need to go out and buy a golf ball.) The largest difference in energy levels will be between the 1st level (n = 1) and the "infinith" level (n = infinity.) So find this difference to demonstrate that (realistically) the energy difference is too small to measure. (The energy spectrum has an upper bound, so what is ?)