1),2) Use sympy
from sympy import *
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
w = Symbol('w')
u = Symbol('u')
q = Symbol('q')
e = Symbol('e')
f = x**3 + x*y**4 + x*z**2*w + u*q**2*w*e**3
f2 = (f*f)
F = integrate(f, x)
G = integrate(f, y)
3) Curve fitting is a different problem. I suggest you look at scipy.optimize
module.