It is important to disambiguate some concepts first. LiDAR data is point cloud data. It can be of types binary or non-binary, but it is not LiDAR data you are working with.
Instead, you have a raster, which is roughly a regular grid having one variable/information represented inside each cell (pixel) of the grid. DSMs and DTMs can be derived from LiDAR data, and they represent surfaces, where the information inside each pixel is elevation data (the z values), and the grid is georeferenced, in which every cell of this grid has a x and y coordinate. Here is a GIS post to understand the differences between DSM and DTM.
The type of raster file you have is .asc, which is also known as ESRI ASCII raster format. Here is another GIS reference about .asc files. Now, let's go to your questions:
I need to know what a single point is (i.e. what is x, y and z).
The first pixel (origin) from your raster, which is the most top (or bottom) left of the grid, have x and y coordinates corresponding to the xllcorner and yllcorner coordinates of the .asc file, hence 350000 and 300000, respectively. These are the coordinates from the lower left corner of the first pixel (if it was the center coordinates, then, the .asc file would carry xllcenter and yllcenter instead). Then, the z value of the first pixel is 118.700.
Another thing I would like to know, is how to actually use the data to pick a location... I'd like to be able to know what areas, specifically, that the numbers are defining.
You can directly import .asc files in almost all GIS software such as ArcGIS, QGIS, R, etc. To know its location open the file together with other vector data as background reference (such as a map from UK, for example). Note, you need to know in which coordinate reference system (CRS) your raster is, in order to get the exact location. If you don't know what it is, this post might help.
Best Answer
You already have a DEM; there is no need for you to create one. The DEM is contained within your files, i.e. you have two copies of the DEM, one contained within an ArcGIS ASCII raster and the other within a GeoTIFF. These are simply file formats that contain the raster data that is your DEM. One of the most common formats for a terrain model is as a regular grid (raster) in which each cell in the grid contains a single elevation value. When I imported your Arc ASCII file into another GIS (I don't have ArcGIS on my machine) this is what it looked like:
I suspect that this is what you refer to as 'a grayscale flat plane', but I assure you that this is a DEM. If I change the colour ramp (palette) used to render the raster and combine it with a hillshade image derived from the elevation data, this is what it looks like:
Granted this is one of the most boring DEMs that I have come across, having worked with DEM data for my entire career, but it is nonetheless a proper DEM. It's just that there isn't much interesting topographically speaking about the data. Also, I can tell from the hillshade image that it was interpolated through a TIN algorithm, which may not have been altogether successful in this particular application depending on exactly what it is you want to do with the DEM.
If you have multiple DEM tiles like this one, then you will need to stitch them all together into a single raster, a process that is known as Mosaicking. Then what you do with your mosaicked DEM depends entirely on your particular application. Perhaps you want to extract stream networks or watershed. Maybe you want to use it to model contamination of surface waterbodies or to measure the spatial pattern of landscape visibility. Maybe you just want to create a cool looking 3D visualization of the terrain...something that you could use the ArcGIS 3D Analyst for if you have the licence for that. There is a vast number of things that DEMs can be used for...welcome to the exciting field of geomorphometry!