Should I be able to use [[+searchValue]] to access the search text from within the noResultsTpl? Keeps returning an empty string; should I be using something else instead?
Yes, I guess so, try to use [[+query]] instead of [[+searchValue]] in the noresults chunk!
Looks like the placeholder is different if results are found or not:
found: [[+text]] (untested)
not found: [[+query]]