What is empty space?
What kinds of energy are in empty space?
What is the process by which the raw energy of space condenses to form matter?
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3 different questions but let's give it a try, since they are related.
Empty space, or really empty spacetime is space, or spacetime with no matter or radiation particles, and in general relativity (GR) that means a stress energy tensor of zero in that region of spacetime. Still, it may have some vacuum fluctuations, see answer to 2.
Quantum theory tells us that the value of different quantum fields, even if zero on the average and in their ground states, will have fluctuations. Those fluctuations have fluctuations as do most quantum entities, they come and go, and last a small period of time, with the time durations roughly given by the Heisenberg's Uncertainty Principle (HUP) that relates the energy and time variances. It is believed that somehow (but this is not fully figured out, still a mystery) those quantum fields, there even in vacuum, are somehow responsible for the Dark Energy that is the cosmological constant that has a very small but cosmologically relevant energy contribution to the universe.
We do not know how the numbers are what they are, but that Dark Energy contributes about 70% of the universe's energy, yet we still don't know exactly how.
No, the vacuum energy of space or spacetime does not condense to form matter or other forms of energy. It seems to be staying constant during the universe's evolution. So what forms matter: it is believed to have come with the universe, except in a different form. A few instants (see the reference below for when) after the Big Bang, a quantum field which is called the Inflaton field started decaying down to its ground state and releasing all the energy it had stored up into a large amount of different particles and radiation, from which eventually came the elementary particles like electrons, gluons, quarks and other things, that formed the matter and energy we see today.
Inflation was invented years back to explain some other issues in cosmology (mostly why the universe is so homogeneous and isotropic) and is based on some Inflaton field which has still not been nailed down. It is accepted as part of modern cosmology, but with some trepidation and purist observationalists will say we can model the universe from then on. But theoretically and some observations point to something like inflation.