I’m fascinated by it, and I’m thinking of making it my concentration. (Linguistics major, CS minor, then a Comp Ling MA.)
Tell me everything there is to know, including the viability of my proposed academic track.
Do you read xkcd?
I’m in computational semantics.
All you need to know is that you should never trust anyone who wants to break an ontology into pieces. That means they think they should be using a visual tool, and that means they’re a nitwit.
I’m a grad student in CS, focusing on natural language processing, so basically I’m on the applications side of things. There’s a lot of machine learning and statistical methods involved, I actually have very little training in pure linguistics.
What would you like to do with an MA in comp ling? FWIW, with your background you could also apply for MS CS programs.
Actually, I have that one bookmarked. 
I’m currently in the final phase of an undergradute degree in computational linguistics (actually “Computerlinguistik” which is more or less the same thing but a bit more inclusive.) I don’t really know what I could tell you. What do you already know or what do you expect? Perhaps I can answer some questions, as long as it’s doesn’t require any knowledge of the US college system.
It’s difficult to say anything in general because even within this niche there are still many different subfields and approaches. There’s a lot of truth in that comic but of course I don’t see it that negatively. 
How big a role does Chomskyian generative grammar play in computational linguistics, really? (See the alt-text for the comic linked above.) Of course, I’m an unabashed Chomskyist myself, so I’ll talk about generative grammar and the language acquisition device all day long!
Other than speech recognition, improved search engines, and cars that talk to their drivers, what practical applications/jobs exist in the computational linguistics (or Computerlinguistik 
) field? Will speaking multiple languages have any real impact on my employability?
It seems that a Computer Science minor won’t have me taking any math higher than this. Will that handicap me in my job search or my Computerlinguistik (sorry, I love that word!) career?
Generative grammar in the general sense certainly plays a role and Chomsky had a great part in shaping the idea of a modern grammar theory. However if you are talking about Transformational Grammar (and its various updates) then at least here it is usually rejected as useless for computational applications. Rumor has it that some American departments are more Chomsky-friendly.
That takes us back to the criticism in the comic. It’s a general problem that natural language is a huge, complex and amorphous beast. We do not know The Truth[sup]TM[/sup] on how it works and probably we never will. All we have are idealized and simplified models. Some of these are good approximations of reality and very useful in practice but only the most die-hard fanatics would claim that any single one is the only correct answer. In real life you have competing and, yes, sometimes contradictory approaches. Some are so severely flawed that they are abandoned but often it’s not a question of binary right or wrong, just degrees of usefulness.
“Improved search engines” in a slightly broader sense is a vast field. You can subdivide it into areas like “Information Retrieval” - finding existing documents that match certain criteria, “Information Extraction” - retrieving structured information from potentially unstructured sources and Question Answering - answering arbitrary natural language questions with generated natural language answers. Beyond your typical search engine website all of those can be intregrated into all kinds of other systems. e.g. the NSA computers that monitor this post (Hi!) are just a different a variant of the same overall idea of making natural language information available for automatic or human use. Machine translation is another major application. Many, many things aren’t really end user products but tools and resources for making these products like parsers, grammars, corpora…
That’s of course a bit different for a native English speaker but knowing additional language is always a nice thing. As long as you stay in the English-speaking world most projects won’t require anything else. For educational reasons we were strongly encouraged to take courses in a non-indoeuropean language, usually Chinese or Japanese (my choice) but that was only for broadening our horizon, not enough for any practical use.
I don’t think the level will be a problem. I don’t know what would be covered besides the course that you linked to but you want at least some statistics. All in all the field isn’t all that math-heavy unless you want it that way.