This really wasn't as bad as it looks. I got:
1.) 1/12 = 8.333%
2.) 4/(15*sqrt(3)) = 15.396%
3.) 0
Explanation for 1.) The polynomial x^2 + Ax + B has two real roots when A^2 - 4B >= 0, which comes from the discriminant part of the quadratic formula. If you look at the graph of the function B = (A^2)/4 in the A-B-plane over the square -1<A<0 and 0<B<1, the region where B < (A^2)/4 has an area of 1/12, which you can get by integrating.
Explanation for 2.) The polynomial x^3 + Ax + B has three real roots when -4A^3 - 27B^2 >=0, which comes from the discriminant in the cubic formula, which I had to look up because I am not among the elite who have that one memorized, LOL. Using the same technique as the previous problem the integral B = (-x)^(3/2) * 2/(3*sqrt(3)) over (-1, 0) gives you the answer.
Explanation for 3.) Polynomials of the form x^n + Ax + B have at most 3 real roots when n is odd, and at most two real roots when n is even. You can use derivative calculus to see that these curves do not have enough direction changes to cross the x-axis very often, nor can they have high multiplicity roots at x = c, for some real number c, due to the restricted shape of the function. You can also look at the intersection of the curves y = x^n and y = -Ax - B to see that you can't get more than 3. So, if n is greater than 3, there is a 0 probability of the polynomial having n real roots.
Not a bad set of problems.