This is a common point of confusion, not only with regards to inflation, but any time an expanding universe comes up...
The "cosmic speed limit" as you call it says that no particle or signal can move through spacetime faster than the speed of light. Spacetime is a very specifically defined thing, described with a coordinate system. There is no restriction, in terms of speed, on what spacetime itself is allowed to do. Let me illustrate with an example.
Imagine a photon. Relativity tells us that it always travels at speed $c$ (exactly at the speed limit). Let's say the photon has a path 10 light years long to travel along (remember light years are a measure of distance, $1\mathrm{ly} =$ the distance travelled by a photon in 1 year). The photon leaves and travels for 5 years, covering a distance of 5 light years. Then very suddenly, the universe doubles in size! The photon continues on its journey. The 5 remaining light years to travel have doubled in size, so it travels 10 more years to cover the last 10 light years. The journey has lasted 15 years. But the photon is now 20 light years from its starting point. Naively, we might compute its speed as $v = 20\mathrm{ly}/15\mathrm{yr} = \frac{4}{3}c$, faster than the speed of light. But in reality, it was just moving at speed $c$ the whole time in a universe that expanded.
In a more realistic scenario, the universe doesn't "suddenly" double in size, it does it gradually, but conceptually the same thing happens... you just need to use integrals to work out the math.
As to the meaning of time, that's somewhat more philosophical. However, I'll point out that, at least in general relativity, time is on an equal footing with space. Spacetime is described by a mathematical object called a metric. One example of a metric looks like:
$$ds^2 = c^2dt^2-dx^2-dy^2-dz^2$$
$x,y,z$ are the spatial coordinates and $t$ is the time coordinate; $s$ is a sort of generalized measure of spacetime length. As you can see, other than the constant $c$ (which could be set to equal 1 with a clever choice of units, so it's really rather unimportant), and a negative sign, time and space are equivalent in this formalism. If you understand space, then time should also make sense, as it's simply related to space by your "cosmic speed limit".
In Newtonian mechanics, a particle might gain kinetic energy while a corresponding gravitational potential energy decreases, thus you get that kind of conservation of energy. The total energy is the same before and after any event. However, the amount of energy depends on who's looking.
In Special Relativity a transfer of energy has to happen at an event (a specific time and specific location), so you have to have changes in kinetic energy be compensated by a loss of energy in another body or field. An example is the electromagnetic field which has an energy density, momentum density, and stress at every point in spacetime. Energy can transfer from the electromagnetic field to the particle and thus you get conservation of energy. It can be expressed by saying the energy in some region of space at some time is equal to the energy at an earlier time plus or minus the net flux of energy in or out of that region of space during that time interval. But energy conservation is just one part of a unified energy-momentum conservation, and that conservation can be expressed in a frame independent manner.
In General Relativity you generalize the kind of conservation of energy as is found in Special Relativity, but the tensor T, called the stress-energy tensor, that keeps track of the energy density, momentum density and stress at every event in spacetime is actually the same tensor from Special Relativity, and so it has no terms that correspond to gravitational potential energy. Breaking the stress-energy tensor into just an energy part is frame dependent and General Relativity is formulated in a frame independent manner. Some people try to make an energy psuedo-tensor, but that is a different tensor. And it is the stress-energy tensor T (not the psuedo-tensor) that is the source of the gravitational curvature, just as charges and currents are the source of electromagnetic waves.
So simply put, don't expect General Relativity to have something like "total energy of the universe", because that's just something that isn't naturally there. There is a stress-energy tensor, which if you pick a frame gives you an energy density at an event, but there is usually no natural frame, so no natural energy density.
But when talking about the stress-energy tensor in different epochs, there might be a sense where is has certain properties and at other times has other properties. One property a stress-energy tensor can have is whether it satisfies various so-called energy conditions. And a common consequence of many energy conditions is that the energy density (for every frame) is non-negative. So the question about whether a particular stress-energy tensor has a negative energy density is a legitimate question in General Relativity.
Best Answer
This is called vacuum fluctuations as stated by Quantum Field Theory. In this case energy has to be supplied so that the electron positron pair out of the vacuum would materialize. This does happen at laboratory energies in experiments with strong laser beams . It is also supposed to happen with the energy supplied by the gravitational field of black holes, one lepton falling in the black hole the other leaving: Hawking radiation
You must not have read the article you link to the end:
So in the active research front of cosmology model spontaneous gravitational creations are being proposed.