Most likely the .TSK file and the .GTO file are the same in purpose. Also, it is quite possible the TI files have no silkscreen on the bottom layer hence there is no file provided.
As there is no standard for Gerber file extensions, there is also no standard as to what files are sent to a PCB manufacturer. The TI files, for example, come with a .BRD board outline, but this is not always a requirement and it is a file you don't seem to have from Eagle (though you can easily generate it by modifying the CAM file to generate an output based on the Dimension layer).
Same with the .FAB file...who knows what is in it? Various fabrication information that is optional and could also be communicated with a text file perhaps.
You can view the Gerber files yourself using a previewer such as GC-Prevue, ViewMate, or gerbv (my favorite) under Linux/Mac OS X. Then you can see what is in them and match them up with what you expect.
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