Will a 9-speed Tiagra shifter work with a 9-speed cassette and a 10-speed Ultegra derailleur?
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:


ST-4400: http://techdocs.shimano.com/media/techdocs/content/cycle/SI/TIAGRA/ST-4400/SI_6CT0C_En_v1_m56577569830607060.pdf
ST-4500: http://techdocs.shimano.com/media/techdocs/content/cycle/SI/TIAGRA/ST-4500/SI_6LP0A_En_v1_m56577569830607086.pdf


Apparently it's been designed so that with two chainrings instead of three, the levers click twice instead of once.