Yes, it is. Lightroom can do all the things you asked for (and more).

The process should be non-destructive and the images unchanged unless I explicitly allow this (see points below).

Lightroom does this. All the adjustments you make are stored in the Lightroom database, the original RAW files are never touched in any way (though you can chose to convert proprietary RAW files into DNG, but this is optional). It may overwrite your accompanying JPGs, but only if you tell it to export JPGs into the same folder where the RAW files reside and it will ask your permission before overwriting any files.

The images should remain in their current location and not duplicated elsewhere.

Upon importing, you can choose whether to leave the pictures in their current locating, or to move or copy them to a new directory of your chosing. Make sure to select Add in the import window to have them remain in their current location. This way, you will also be able to browse your photos using your existing folder structure inside of lightroom.

I would like to be able to add geotags to the EXIF data in the images, via GPS tracks or by using a map on multiple images where a GPS track isn't available.

Lightoom has a Map module for this purpose. I have never used it, but you should be able to do anything you want with it. See here for using the Lightroom map, here for an introduction to Geotagging and here for an in-depth introduction to the Map module.

I would like to be able to tag people in the EXIF data by using facial recognition.

Lightroom has an automatic face recognition that you can turn on for all imported photos or only for selected folders. You can also manually delete recognised faces or add new ones. Lightroom will try to learn faces and suggest a name for recognised ones, you can either confirm or override the suggestion.