1 I can't remember since when computer engineering has DSP course.
2 DSP does no programming, you are going to be thrown into a pool of math, it prepares you for future research work.
3 Most likely you will not be touching signals at all because you have not enough math fundamental skills to talk about digital signals.
4 If you are not going to specialize in digital communication, do not choose it.

neverthless, the course itself opens a door for you, although you are only given a glimpse of digital communication world.