My son's race bike currently has a mixture of components... A 9-speed 14-25 cassette (juniors race cassette), Tiagra shifters, 105 rear derailleur, Tiagra triple front derailleur and Tiagra triple crankset. He would like to upgrade to an Ultegra derailleur, compact crankset,and brakes that he can get used. Will the Ultegra derailleurs and brakes work with the Tiagra shifters? He will be sticking with the 9-speed cassette, so I suspect the rear shouldn't be an issue. Any insights would be much appreciated!
In general, parts should be compatible between Tiagra, 105, and Ultegra.
As you mentioned, the 10-speed rear derailleur should work with 9-speed shifters:
Within a given brand/style of rear derailer, all "speed numbers" are generally interchangeable. This applies to all indexable models, basically everything manufactured since the late 1980s. There are a few exceptions. -- Sheldon Brown (6-speed, 7-speed, 8-speed, 9-speed, 10-speed, 11-speed?)
However, switching from a triple crankset to a double crankset will be a problem unless you also replace the front shifter with one that is designed to work with a double front derailleur.
Edit: It looks like the Tiagra ST-4400 and ST-4500 are actually compatible with both double and triple front derailleurs:
Apparently it's been designed so that with two chainrings instead of three, the levers click twice instead of once.