One answer. You've asked whether to use drop-downs. First, when I looked at the illustration you posted, I didn't recognize your boxes as drop-downs because there's no ▼ cue that anything will drop down. I suppose you could add a glyph (triangle) to signal that these boxes is a drop-downs, but there's a problem with that: drop-downs are unusable for a general audience—at least in the UK. To back up this statement, here's Alice Bartlett's presentation on banning select tags, which is an excellent talk. Highly recommended to watch the whole video. The video also points to NN/G article, Drop-Down menus: Use sparingly, and Luke Wroblewski's article, Dropdowns should be the UI of last resort.
Another answer. Here's my own anecdotal experience. I solved a very similar design problem in a very similar way as your solution. We released, and then I looked at the search logs. Surprisingly, to me, the log showed the site's registration-related search results were performing poorly. (For obvious reasons, I'm not naming the site.) Parents looking for courses for their kids didn't seem to be able to apply the correct set of filters/facets, and even when they were able to, the log showed they didn't look past the first few results, so they missed most of the course choices. And I don't think this is a one-off result. Do you remember the Google search page of a few years ago? it used to have a few search refinements. Then those disappeared, but it still had a link to Advanced Search. Now, the only way to find Advanced Search is to Google it in the regular search. In general, people don't use filters/faceted search, and Google knows it.
On the bright side, you've got a product you can test, you can set a baseline against which to measure future improvement, and you have an opportunity to improve. "Yay." Right?