A "Mathematica script" means something specific. Are you looking to write a program that runs in Mathematica or externally?
1. See the documntation for Manipulate on how to make simple user interfaces. You can make user interfaces that take an inputfield. For example:
Manipulate[Plot[f[x], {x, 0, 10}], {f, Tan}]
2. For example:
Manipulate[Integrate[x y, b], {b, x}]
3. This really wouldn't be much more of an addition to the previous example. Maybe it could be something like:
Manipulate[With[{firstResult = Integrate[x y, b]},
Integrate[firstResult, z]
],
{b, x}]
4. You can use Export to export something to a file as a specific file format. In this case, you'll probably want a button that causes an export to happen. See this example from the documentation in Manipulate on how you can make a button:
Manipulate[x, {{x, 0}, Button["random", x = RandomReal[]] &}]