If the data is in GCS_WGS_1984 you will find the cell sizes in the properties to be a little confusing, as well as having quite small values. This is because the units are in decimal degrees, not meters/feet. For example, I have some 30m ASTER data for the world, but when I check the properties the cell size is 0.00027777778, which is due to the data being in GCS.
If you plan on analyzing the dem, you should project it such that the linear units are the same as the vertical units so that X, Y, and Z are consistent, otherwise expect to be calculating Z factors and using them when appropriate. Note that Z factors change with latitude so if your extent is very large a single Z factor won't cut it.
When projecting a dem, do not accept the default resampling in the Project Raster tool. Switch it to bilinear to save yourself some headaches down the road. Nearest resampling is not meant for a dem.
Best,
Eric