But study results are only published if the study is done, and the study is often only done if it gets institutional approval for the proposal.

Yes, but a study's conclusion isn't preordained. It's entirely possible for authors to end up disproving their hypothesis. So it doesn't follow that, because more research proposals about "reverse discrimination" were rejected, then more studies concluding "reverse racism" (holding all else constant) would be rejected.

Yes, both are covered. Studies that specifically look for inconvenient facts don't get funded, and studies that accidentally find them don't get published.