Is dark energy homogeneous and/or isotropic? Shouldn't there be at least some fluctuations of dark energy in various regions, analogous to the fluctuations we observe in other forms of energy like matter?
 A: Both the isotropy and homogeneity are amenable to observational test. One can measure the peak brightness and redshifts for type Ia supernovae in different directions and at a range of redshifts. One can then see whether the same cosmological models (including $\Lambda$) are required or can consistently model all datasets, or whether there are angular correlations at particular scales in the peak magnitudes of supernovae at a given redshift.
There are actually quite a lot of papers that have examined this question - some have claimed to detect an anisotropy (at low significance) - e.g. Cai et al. (2013), whilst others find no such signal - e.g. Blomqvist et al. 2010. 
Another possibility is to look for the signature of dark energy inhomegeneity or anisotropy imprinted on the cosmic microwave background (e.g. Weller & Lewis 2003).
A review of this plus some other suggested diagnostics - Baryon Acoustic Oscillations, Weak Lensing measurements etc. are reviewed by Battye et al. (2015). My (very) limited understanding of the situation is that inhomogeneities would be expected if the dark energy obeys an equation of state that is other than the $P = -\rho$ corresponding to a cosmological constant, where $P$ is the pressure and $\rho$ is the energy density due to dark energy. 
In other words, if $P = w \rho$, with $w \ne -1$ (for example in dynamical scalar field theories for the dark energy such as Quintessence, or k-essence), then clustering of dark energy is predicted
The final conclusion of Battye et al.'s review is that any clustering in the data is limited to scales larger than $30 h^{-1}$ Mpc, and the data are still compatible with no inhomogeneity and $w=-1$.
A: Yes, if the extra curvature is due the energy density of some unusual stuff that we just haven't seen yet, then it is reasonable that the density would be different at different places and times in the universe.  
However, if experiment shows the extra curvature to be the same everywhere, then it may be a new physical constant ... and not due to the density of any stuff.  Curvature has the dimension of [1/Length^2] so it would supply a new fundamental constant of length. This is kind of like the discovery in ~1900 that the speed of light is a fundamental constant in free falling frames everywhere. Such a new fundamental length constant would stimulate all sorts of new theoretical ideas.
