>Suppose that the early universe was filled with energy uniformly. Would quantum fluctuations cause variations in energy density across space resulting in regions with higher and lower energy density around the mean?

The inflation period and the inflaton field were proposed in order to explain the high uniformity of the cosmic microwave background, to a level of 
$10^{-5}$, among other observations. The small inhomogeneities observed below that leve, are explained by [quantum fluctuations][1] at the time of the inflation period.

At the time of inflation, there are no observers, only interactions of fields. The inflation model assumes perturbations in the metric (see link above) at that period . The [wikipedia article][2]  on cosmic inflation states 

>It explains the origin of the large-scale structure of the cosmos. Quantum fluctuations in the microscopic inflationary region, magnified to cosmic size, become the seeds for the growth of structure in the Universe 

it refers to [quantum fluctuations][3] as 

>quantum fluctuation (or vacuum state fluctuation or vacuum fluctuation) is the temporary change in the amount of energy in a point in space, as explained in Werner Heisenberg's uncertainty principle. 



The map of [cosmic microwave backround][4] is the "measurement" of the fluctuations at the time of inflation. Certainly space time is perturbed during inflation and the cosmological constant allows for energy fluctuations, which are reflected in the CMB map.


  [1]: https://www.mpi-hd.mpg.de/lin/events/group_seminar/inflation/schmidt.pdf
  [2]: https://en.wikipedia.org/wiki/Inflation_(cosmology)
  [3]: https://en.wikipedia.org/wiki/Quantum_fluctuation
  [4]: https://en.wikipedia.org/wiki/Cosmic_microwave_background