Log into the web UI and navigate to Management > Disk Management > S.M.A.R.T. Test. The S.M.A.R.T. Test will perform a HDD diagnostic and let you know if the ShareCenter finds anything wrong.
You may also want to go to the HDD manufacturer site (e.g. Seagate, WD, etc.) and download/install the HDD test tools on your PC. Since these tools are designed for specific make/model HDDs, they may provide a better indicator as to whether your HDDs are healthy.
If you have a HDD that is failing and the HDD is still under warranty, you can request a replacement from the manufacturer.
Regarding your specific questions, I don't have all the answers, but here's what I can offer:
(a) Unhealthy means that the DNS-345 has determined there is something wrong with the HDD. I can't say whether "Unhealthy" refers to the HDD's physical health (e.g. mechanical) or logical health (e.g. data corruption, filesystem issue). The S.M.A.R.T. test will look at the physical health of the HDD.
(b) I'm not sure what circumstances would cause erroneous results (run the tests I describe above)
(c) I don't know if the front LED will provide any indicator messages regarding HDD issues