calculate river slope from coarse DEM

Here is one way to calculate the slope of a stream (and long profile) from a coarse DEM.

(1) If the DEM is huge, clip it to only the area of interest, to speed up the processing. You can do this by creating a polygon outlining the area, adding it to the map, and then using extract by mask (Spatial Analyst, Extraction, Extract by Mask).

(2) It may help to visualize things by making a Hillshade.

(3) Under Spatial Analyst, Hydrology, there are three steps.

(a) Fill

(b) Flow direction

(c) Flow accumulation

(4) Now you want to extract the stream by choosing an upstream point, then selecting all points with a flow accumulation greater than that value, using the conditional to turn the map to 0s and 1s to delineate the stream, and then turn that stream into a vector.

Map algebra, Single Output Map
Conditional

Hydrology
Stream to feature (may be optional and affect choice in Step 5)

(5) Select equally spaced points along this line either by using Hawth’s tools / Geospatial Modeling Environment or this previous post.

(6) Extract Values to Points, Export data

(7) You can open the .dbf in Excel or Gnumeric to inspect the slope.

There’s also this:
Tarboton, D. G., R. L. Bras, and I. Rodriguez-Iturbe. 1991. On the Extraction of Channel Networks from Digital Elevation Data. Hydrological Processes. 5: 81-100.