The DEM tiles are already projected if they are in NAD 1983 UTM Zone 17/18N. From what I understand, a Mosaic Dataset IS able to handle multiple projections: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//009t0000000n000000 just search this page for "different projections" and you will find a brief paragraph explaining this.
You will, however, have to specify a co-ordinate system for the Mosaic Dataset, which would probably end up being GCS North American 1983...just go to the properties of each raster, and where it tells you the 'Projected Co-Ordinate System', scroll down a bit to see the 'Geographic Co-Ordinate System'. The Geographic Co-Ordinate System should be the same across all of the rasters, so whatever that is that's the one you will want to use for the Mosaic Dataset. More about that here: http://blogs.esri.com/esri/arcgis/2013/01/16/how-does-a-mosaic-dataset-handle-rasters-in-different-projections/.
Then, in order to do your analysis, you will probably have to project the Mosaic Dataset a few different times depending on the layers you are creating due to the distortion that is associated with different projections. Read this resource: http://mappingcenter.esri.com/index.cfm?fa=ask.answers&q=435.
So, to reiterate:
- Your DEM tiles are already projected
- A mosaic dataset can handle the different projections...it is basically projecting each tile 'On the Fly' into the Geographic Coordinate System that you specify
- In order to work with the mosaic dataset, you will have to project it so that it also has a Projected Co-Ordinate System as well...remember this will have to be done a few times for different distortions/outputs
You will have to do some reading about how layers such as slope and hillshade are derived, as well as projections and distortion. The last link I included in this answer is a good starting point to understand this concept. From there, you should be able to find resources on Google.