Calculate pH of a buffer solution
Is the answer key correct?


Yes, they have a very good approximation of the pH, and their way of solving it is straightforward and conventional.


  After doing a BCA-table, I got that the remaining moles of HCOOH = 0.003moles


The sum of formic acid and formate has to remain 0.027 mol. You could start out with 0.027 mol of formic acid and turn some of it into formate by adding NaOH. Or you could just mix them as described in the problem. However, you are claiming that there is a total of 0.015 mol of formic acid plus formate, so you lost some atoms there. As the comments state, there is no reason to think about where they got the sodium formate (stock room?).


  If the answer key is correct, why does the pH decrease?


Decrease or increase depends on your frame of reference. If you compare it with a 1:1 buffer where the pH = pKa, it decreased because there is a higher concentration of acid than conjugate base.

If you compare it to the pH of 0.3 M formic acid (roughly pH = 2.1), then it increased because you added conjugate base (neutralized and diluted the acid).