Most any resistor of 5K ohm will work, if max amps flow (assuming the photo transistor is a short) the current would only be .0006 amp , and wattage used would be 0.0018 w (1/18000) - I'm not aware of a resistor with that small of a wattage, so about any 5k you can get will work. The resister in photo appears to be 1/4w or smaller.
Am skeptical about any phototransistor working however. The phototransistor chosen should have a rise time shorter than the shortest desired measurements. To measure 1/1000 s one would need at least 1/2000 minimum rise time.
There is nothing wrong with testing different phototransistors however, using a camera with a known good shutter (consistent negatives, using all combinations of shutter speed/aperature) will allow you to test the tester.
Of course infrared will work, however, you need to insure that infrared exists in the light source. Probably not from a visible light LED.
Using 9v instead of 2 AAs (3v) would triple the current (probably not a problem) but why not use the 2 AAs as designed. Using a potentiometer instead of the fixed resistor would allow you to adjust to get the output in the desired range, but is a needless complication, use the 2 AAs.
Solder? If you can solder with it, it is the correct gauge. However, do NOT use acid core solder, resin core only. When doing really small work, I tend to use smaller solder gauge than when soldering large items. Using large gauge on small items can tend to oversolder the joints, using small gauge on large items can use an awful lot of small solder.