Cara
you said that the output raster is identical to dem1 and doesn't include values or counts for dem2 and dem3. Do you mean the pixels and extents of dem2 and dem3 were simply ignored and not properly added into the output raster? If that's the case, it sounds like a bug. What version you're using? If you're on an old version, this may be a bug that was fixed long ago. Are you using ArcGIS Pro? ArcMap? You should work with Support on this.
If I misunderstood and the pixels are all there but simply the attribute table has not been updated, have you run "Build Raster Attribute Table" on the new raster after combining the three source rasters?
Last, (re: Dan's last comment), don't confuse a Mosaic Dataset (a database structure unique to ArcGIS for accessing and mosaicking separate rasters on-the-fly) with a mosaic. This is the first mention of a Mosaic Dataset (which I might recommend if you're not using the resulting file outside ArcGIS, and you have some experience with it) but I assume you have a reason for wanting to create a new file with all three original datasets combined into one.
Cody