Where does the energy come from that produces virtual particles and are they related to string theory?

They are entirely described by current physical theories and therefore not the most interesting part of string theory...but yes you could say they are related to string theory - but only in the same way that all sub-atomic physics is.

The energy does not "come from" anywhere in the normal sense of the phrase - it is a consequence of the time-energy uncertainty principle. There is inherent uncertainty on the product of an energy and the time for which it is around. This leads to the interesting result that energy can appear from no-where...provided that it disappears quickly enough. By the maths, the larger the energy, the shorter the time it can exist for.

Relativity shows that energy and mass are interchangeable...and so by a combination of these two ideas we can see that particle-antiparticle pairs can be produced in a vacuum, as long as they annihilate each other quickly enough. We can also simply see that the larger the mass of these pairs, the shorter the time they can exist for. It is photons created by this method ("virtual" photons) which are responsible for the electromagnetic force (in QED).

>ww<

This is the best I can do for a Feynman Diagram of a quark to quark coupling with a gluon as the virtual particle.  The two quarks are >  and  <, with the ww as the gluon.   In Feynman Diagrams, virtual particles are typically dotted or wavy, like the ww, and start and end with a propagator line, like the > and < in my diagram.

The gluon is called a "virtual" particle because it, like all virtual particles, does not get its energy from normal space-time in the same way real particles do.  But the gluon gets its energy and ability to transfer momentum between the two quarks by borrowing from space-time and then giving it back before real space-time realizes it lost that energy for a brief moment.  

Only in the world of quanta where the Heisenberg UP reigns supreme can this be done.  Remember, in HUP, energy and time are the two uncertain factors, as well as the more commonly known location and momentum.

This, by the way, is the same property that results in quantum tunneling.  So if you want more on the borrowing energy bit, you can find something on that capability when reading up on quantum tunneling as well. [See source.]

I almost forgot your other question.  String theory does indeed predict the virtual particles.  Its major claim to fame is that it covers all the particles in the standard model, including the anti and virtual ones.  I've never seen the complete list of predicted particles; that would be interesting to see.  Anyone have any citations for such a list?