When I add DTED data files (*.dt1 and *.dt2) to my ArcGlobe application, they are displayed as solid gray boxes. Do I need to do something special to the files and/or the ArcGlobe control to see these properly?
You really shouldn't be managing individual DTED tiles yourself if you plan on having any substantial amount of DTED data in your application. At ArcGIS 10.0 you can use Mosaic Datasets to store many different types of imagery data including DTED 0-2. Pre-10.0 if you have access to Military Analyst you can use a MA DTED Catalog to store DTED data.
ArcEngine provides you the ability to call any geoprocessing tool (assuming you are licensed for it) programmatically. I don't know which API you are but they are generally the same across the different languanges. There are two methods on the geoprocessor to execute a gp tool. The first takes an array of arguments which you have to figure out via the GP tool documentations. The second provides a bit more safety in that it will give you the variable names of a particular GP tool and let you set them individually.
If you are using Mosaic Datasets then you can add DTED data to it by using the AddRastersToMosaicDataset GP tool:
Merge the DTED dt1 rasters into a File Geodatabase Raster Catalog, then use the Raster Catalog to Raster Dataset tool (in Data Management Tools -> Raster -> Raster Catalog To Raster Dataset) to create one big mosaic. So far so good and it performs well and maintains its sense of elevation... repeat for dt2.