Hello Anna,
I would not replace the notebook if it is only the HDD that has failed, you have been able to run various other types of test and apart from some issue/s with the DVD drive everything else seems to be fine.
Things would be slightly easier for me as I have the required kit but option one for me would be to remove the HDD from the notebook and test it with something like the example here if I didn`t have such a device I would fit into my desktop, doing the latter would require a SATA cable and power supply cable from the PSU to be attached.
Confirming the state of the HDD is 100% what needs to be done atm, based on the tests done and most notably the fact that the drive could not be mounted using Linux the evidence does point to a bad HDD I`m afraid