Yes with your understanding of ArcGIS image tools and supervised classification you could do this assuming: a) the lidar data has picked up some information that is indicative of spawning habitat b) you have known spawning areas to use for training.
Your ideas of using roughness seems good if you can derive that from the lidar. Unless the point spacing is very tight it will be hard to distinguish spawning gravel (1/4 inch to a couple inches depending on your fish size), from finer sediments and mud, but you might be able to use supplemental imagery to help with this.
For the depth, take a look at the Height Above Nearest Drainage workflow described in the reply to this question.
Here is a pretty relevant project - Modeling salmon spawning habitat from lidar PDF and other project info and downloads
And another somewhat related - Spawning lake trout in Yellowstone