Right.

Yes, this is an F transform of the function a. A Fourier transform "switches" the independent variable to its conjugate. Position and momentum are conjugate pairs.

I really don't know why the dx's etc. are written in front of the integrand. But I find that it makes the equations easier to read, personally.

The uncertainty of an object's position is directly related to the fact that the object is being treated as a "wavefunction." A wavefunction is not localized in space. All we can do is find the region of maximum probability of the particle being there, and call this the position. Since this region is never a point, there must be some intrinsic "smearing" of the position of the particle. The same argument holds for the momentum. (There is a way to argue this in the "bra-ket" formalism, but I'll be if I understand it myself.)

Hint: The minimum energy is going to be for T = 0. Now note that

By the way, here's a Latex tip. The command for h-bar is "\hbar" inside the math brackets.

-Dan