Are joke episodes common in anime?
It depends on the series.

For series who have a guided plot and who have a specific direction they're going in, like Bleach or Naruto, having a comedic episode appear once in a while serves as filler, and something light-hearted for the audience to enjoy.  For instance, non-plot-related episodes of Bleach involving Don Kanon'ji were often joke episodes at intent, and Bleach even did an Arabian Nights' episode in the middle of the major Hueco Mundo plotline.

(I would argue, though, that Kakashi's true face became an ascended meme within the Naruto franchise due to the pervasive curiosity that Naruto and Sakura have when they go to visit him in the hospital, and come this close to actually seeing it.)

Less guided shows like Hayate the Combat Butler have joke episodes embedded and running alongside actual episodes to the extent that it becomes tough to filter out the noise from the signal.  However, with it being a gag anime, and it not getting ever serious (in animation), this is not only par for the course, but to be expected.

I would say that it's not common, but it largely depends on what demographic the show is trying to hit, and how they're portraying themselves - either as a story, plot and resolution-style show, or as a gag show.