I think you're thinking slightly over the top in the end circuit. All you need is a photodiode which is the same wavelength as the laser. The photodiode will have a current flowing because of the light shown on it by the laser. Once the light is interrupted, the current drops so you can take the time from when that happens. Also you need to think a little, I believe drag strips have 2 lanes. so you're going to need to have a separate triggering system for each lane, which isn't too hard to implement.
Here's a link to help try to understand photodiodes:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
If I remember correctly, photodiodes have a response time of a few nanoseconds.