To fill the holes in the DEM I would recommend using the elevation void fill function. It does not require you to convert your current DEM into points. You can see how it works using this link: http://desktop.arcgis.com/en/arcmap/10.3/manage-data/raster-and-images/elevation-void-fill-function.htm.
If you use the Fill function in ArcMap you may be able to remove abnormal elevations (outliers), but I'm not sure if this is the right step to tackle the missing data situation. However, as you have missing data, I would recommend you to use Map Algebra write a one line script to change the missing values (which should have a NoData value assigned). In ArcMap's Map Algebra tool, you can do this:
You can change 10,10 and specify the size of the block you want to get statistical information from while updating the missing data. I hope this helps.
It is possible to fill the "holes" but you will potentially be introducing some error.
1. As Dylan suggests you can use the Elevation Void Fill function in ArcGIS (http://desktop.arcgis.com/en/arcmap/10.3/manage-data/raster-and-images/elevation-void-fill-function.htm). Here's also a link to a more detailed explenation of the FILL function: http://pro.arcgis.com/en/pro-app/tool-reference/spatial-analyst/fill.htm
2. Check out the second answer in this post, for a more "hands on" solution were you yourself create a moving window (you decide the size of the window) that checks for NULL VALUES in the data and replaces them with the average of the surrounding windows: https://gis.stackexchange.com/questions/256116/filling-hole-in-dem-using-arcgis-desktop
This way you can control the interpolation parameters and check how much the results change if you apply a larger or smaller window. In other words, you will by able to quantify the error/level of uncertainty you are introducing.
3. Have you checked if there are any older DEM-layers over your area of interest? If this is the case and the elevetion values are not much different (given no hazard has occured to change them drastically), you could perhaps use the old DEM-layer to fill the gaps (cells with a NULL-value) in your current raster. Create a mask from your current raster layer, so that all cells that have NULL-values receive the value 1 and all other values are assigned with the value 0. Then overlay this mask with the old DEM-layer by using the multiplication function. Here's a link with how the overlay raster tool works in ArcGIS (http://desktop.arcgis.com/en/arcmap/10.3/analyze/commonly-used-tools/overlay-analysis.htm). Finally, overlay again the output layer with your current DEM-layer to fill the gaps, but use the sum-function this time instead of the multiplication-function.
You could obtain another DEM over the same region (SRTM 1 sec), convert your existing grid to points and then use co-kriging to interpolate the DEM. This will interpolate your data using the second DEM to guide the interpolation where you currently have nulls.
If you have another DEM (dem2) and would like to replace the NoData values in your current DEM (dem1) with the new DEM (dem2), you could use the following in the raster calculator: