A quick distinction:
In ArcMap, "Point Density" will yield a heat map (mostly used for visualization) while something like "Hot Spot Analysis (Getis-Ord Gi*)" will yield a true statistical hot spot analysis. Here is a link with a bit of information on the distinction:
also this GIS SE post:
Difference between Heat Map and Hot Spot Analysis?
Addressing your error codes:
It may be helpful if you could
- provide the full set or a subset of the data, or
- provide some information about the dataset and environment, ex: how large is your point data (number of rows), are you using a population field to augment the heat map, what scale/extent does your data cover, spatial reference system of the data, data format (shapefile/file geodatabase), etc.
From the information we have though:
000597 – When I see this one, my first step is usually to make a copy of the data (preferably into a file geodatabase) and then run Repair Geometry with the option to remove nulls checked. Just as a sanity check. Then retry.
010429 – This one almost seems like a file permissions error. Are you working on a network drive? Do you have permission for the output location for the heat map raster? Is there enough disk space there?
101244 and 010067 – These are unfortunately vague but perhaps have something to do with the spatial reference of the data. I would recommend re-projecting to a projected system if it is not already in one. I believe an equidistant projection would do well for the neighborhood distance calculation in the Point Density tool.
Moran’s I approach:
Not sure as to the result you are expecting here. May need further information and this may end up needing to be a second question, e.g. “Moran’s I Tool Hanging up with the Following Inputs When Trying to Calculate Spatial Autocorrelation.” I will mostly stick to the heatmap question here for now.
Suggestions for smooth “heat map” (not hot spot, see above distinction):
You can alter the output cell size in the Point Density tool to your desired look. Obviously a smaller size will end up taking a longer time to run. Also, may be a good idea to again make sure you are in a projected spatial reference system with easy to understand units. I find that a geographic SRS makes for a difficult cell size estimation and could even potentially be leading to some of your errors if the input is out of bounds.
Hopefully this helps give you some avenues to research and test. If you can, edit your post or comment to add additional information about your inputs as I mentioned above and perhaps we can assist further.