1.) Yes, if the wavefunction does not have complex parts, then psi*psi is just psi^2. Otherwise, to get psi*, you replace all "i" with "-i". Also, usually when you are looking at observables like the kinetic energy, there will be a "psi*psi" when you find expectation values for them. This usually gives you an i^2 and rids you of the complex number. Which makes sense physically, you don't want an "imaginary kinetic energy"
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2.) Be careful, the 0 to a boundary sound like particle in a box. These are subject to change when the problem changes. But yes, you always will want to work with a normalized function, so if its not already, you will need to normalize the wavefunction before doing anything else.
3.) Yeah, that's a really common mistake so don't feel bad, hopefully your professor wasn't too hard on you. If you get a complex operator like x(d/dx), you want to do the operation that is immediately to the left of the function first. So for x(d/dx)x^4, you would take the derivative first and then multiply by x.
Hope this helps some.