If you run the Check Geometry tool on your file geodatabase feature class (and that is the correct term, feature class, not shapefile, in this case) you will find that the first feature OBJECTID = 1 has null geometry. It is not a valid feature. I was able to check out to axf with an error. The axf only contained the second feature. I was able to edit the feature and confirm that ArcPad registered that it had been edited by inspecting the axf file in ArcPad Studio. On trying to check in those changes, there was an error that crashed ArcCatalog and those edits were not reflected in the feature class.
After editing and removing the first feature, I tried to the same process again and met the same results. Error on check out (Full path to axf string is empty), axf generated, edited, check back in, error, edits not checked in.
In both cases the resulting axf file did not have a layer definition for the feature class, although one is always generated by default.
If I were to diagnose the problem, I would say that you simply have too many fields in your feature class. Since I did specify a full path and name for the axf file despite the Check Out tool reporting an empty string, it makes me think that each one of your fields may be entered as a parameter in the execution of the tool. At some point a maximum size or number of parameters is hit, the tools spits out what it can and returns an error that may be true but is not quite correct.
The maximum number of fields in a SQL Server Express 2005 table is 1024. You have not exceeded that limit but you may have exceeded the capacity of ArcGIS to process the request that Check Out tool is making.
All testing I did was in ArcGIS 10 and ArcPad 10. I tested both with the file geodatabase at the 9.3 version and at 10 version.