Spotlight and finder have stopped finding my files + Time Machine has stopped working as well.

Hello everybody,

A couple weeks ago I noticed my iMac initiated a indexing of my HD, and ever since, spotlight (and finder search cmd+f) have stopped finding some of my files, which is really enoying as I am now losing a lot of time looking for them manually. For example, see screenshot below:

Spotlight couldn't find Disk Utility (nothing is unchecked in spotlight prefs.), though it's there in my Apps folder.

Another example, using cmf+f this time.

I looked for "heures sacha baptista" and Finder could only find 2 files (right), but manually (left), I found 9; see below :

Coincidentally, ever since that re-indexing, my time machine has stopped working as well... Now, when I initiate a backup, it says "preparing" for about 5 seconds, then suddenly stops, and nothing is happening.

Please, does anyone have any idea what's going on here?

Thanks a lot for your help and your time!