I don't think I was able to find all the scans needed for what's in the original posts, but, here.
Dream Manip/Subjective Reality: It's stated by both the
Nepedia description for Heart Dimension and
Kurome herself to be the case.
Mind and/or Empathic Manip from Kurome: Because of Kurome trapping them in the 'whirlpool of delusion',
the CPUs want to kill their sisters and friends and Purple Heart, after being snapped out of it,
questioned why she loathed Nepgear and even said that
she (Purple Heart) didn't want to hate her (Nepgear) anymore.
Omnipresent Kurome in Heart Dimension: Statements about how Heart Dimension is a
dream and
creation of the sealed away and sleeping Uzume are mentioned, as well as Nepgear saying
Kurome's soul is connected to Heart Dimension, with S-Sha shortly after saying that
Kurome is one with the Heart Dimension, essentially saying that she herself
is Heart Dimension, which would give her Omnipresence there.
Acausality (only Type 1): At the beginning of the second arc of Megadimension Neptunia, Kurome was able to rewrite Gamindustri
and its history, Histoire reiterating later in the game that
Gamindustri's history was indeed altered. However, the CPUs' memories were unaffected by it, as well as Compa, a few random citizens, Histoire, the Gold Third, and shortly after that, IF regained her memories as well.
Side note: You could argue that this is a resistance to Memory Manipulation, but given what we see at the moment, I don't find that to be the case.
Resistance to Conceptual Manipulation: The description of Next Purple's 'Dimensional Slice' attack states that it 'slices through even the concept of the opponent', which is in tandem with their resistance to it because of Kurome saying that
the newly created Dark CPUs absorb their powers and recreate them, which would include Next Purple's Dimensional Slice. The cast had already faced off against three of these new Dark CPUs, which would grant them the resistance.
As for the 9-Dimensional Coordinates: There are scenes in which these coordinates
are brought up (albeit, in the last example, the numbers are not displayed nor mentioned, simply that the Heart and Hyper Dimensions are closer to each other), and while it is stated that
the coordinates don't normally change, Croire notes (
near the end of the game ffs) that with Rei Ryghts' power,
moving dimensions is 'a cinch' using only a small portion said power. However, these only seem to serve as a plot factor, not having any real relevance outside of Kurome's goal of wanting to fuse the two dimensions together, so this is debatable as to whether or not it can be used.
But, that's really about it from me. I provided the scans that I could, as I couldn't really find anything else for the others.