Hey there, I can answer your question well, having both attended the MA Film and Media Studies program and worked for the Film and Media Studies department/Jane Gaines (the head of the graduate program).
Insofar as your first question goes, it is by no means as competitive as other graduate programs that don't offer solely a terminal MA at Columbia (or elsewhere) - for instance, the highly-lauded comparative literature department(s) at Columbia are very competitive. You can apply to the Comp Lit PhD and take the terminal MA in Film/Media studies as your "path," so to speak (this is what I recommend doing for the qualified PhD candidate). There's a reason why the Columbia Film and Media studies website doesn't have any numbers re: how many students are accepted a year and how many apply (whereas most grad programs have this within their FAQs page) online. I saw someone a few years ago post (in gradcafe) that 100-150 students apply a year and the program accepts 15; frankly, I have no idea where they came up with these numbers as they simply are not true. The actual numbers are not really widely open to anyone outside the acceptance committee.
Having said that, Jane Gaines is a leading (feminist film/Foucauldian media archaeology) scholar who worked at Duke's literature program, having founded the film studies sub-dept. there. Nico Baumbach, while relatively young, is from Duke's literature grad program (the same program that produced Galloway and Luka Arsenjuk amongst many others). His new book (his first) is excellent! However, as far as actual "graduate level" courses (where you won't be sharing a classroom with undergraduates) go, there are only two-four such classes in the first year (the second year, which is comprised of one Fall semester thesis class, does offer auditing but that is all). One grad level class is the (mandatory) Historiography course Gaines teaches at the first semester and the second is Baumbach's (mandatory) course the second semester (a survey seminar). Otherwise, I would say the program is much more undergrad-oriented. Also, there are TA spots that only go to a few students (I was lucky enough to) but no actual departmental funding, though Gaines does the best she can to take on researchers, rapporteurs, etc. Thomas Elseasser also teaches a mind-game film course in the spring semester (this is one stand-out feature, of course, and Columbia's departmental claim to fame).
To tell you the truth, having completed the program, many more students are accepted or waitlisted than one would expect. Of course, not all, but higher rates than what you may expect from an Ivy League graduate program. As you are free to opt to take other courses, from Art History or Comp lit, for example, many students seeking a "graduate level" experience do this rather than taking undergrad-level courses from Robert King or Richard Peña and Annette Isindorf. While this program is younger than NYU's (the oldest cinema studies program in the country), I would recommend it above other terminal MA programs (including NYUs) but below any PhD program (Ivy League or not).
Lastly, re: what can one say for PhD acceptance? This certainly depends on the student - an MA from Columbia will not grant you much in the way of sure-fire acceptance to PhD programs. The program has yet to place anyone in the graduate program at Duke's literature PhD program, where more or less everyone dreams of going. You will have a 50 page dissertation under your belt, sure, but one could accomplish much of what will prepare you theoretically and such elsewhere (somewhere less expensive and with richer exclusively graduate course offerings for graduate students). No one yet has been accepted to Visual and Environmental Studies (Harvard), MCM (Brown) - other top notch institutions - but they have been accepted to many other prestigious programs (e.g. MCC at NYU). However, there are many students who have been rejected from these same programs. Depends on if you have published work, attended conferences, honed interests, research experience, etc. These will take you much further than the Columbia name, as the Film and Media MA program does not have the high ranking that other departments (at Columbia) have.
Hope this has helped. One last note - if you are interested in comparative media/new media theory, you will find Columbia's program a bit lacking. Baumbach's courses are one exception - he's an excellent resource. NYU's MCC, however, is certainly a more apt option in the area. Consequently, I recommend this program for those who are not good candidates for the fully funded, more established PhD programs and would like 1.5 year(s) to garner this kind of experience first.