In a lot of animes (animii?), humans just have more durability than real life humans. Not sure how I'm supposed to provide sources for this, but in, for example, One Punch Man, Saitama was repeatly beaten down by Crablante before gaining any powers and survived. In JoJo Battle Tendency, Mark gets half of his head sheared off and is still able to talk for a ...
They're called Al-mi'raj. And they have been around for hundreds of years.
Dragon Quest is only the first game to include them in.
Series could just be using them as a starting monster as a simple DQ reference... But it could also be because they're like one the most easiest fantasy animal to kill. The most they can do is scare off wild animals.
In Japanese, a gloomy/depressed person can be referred to as being "damp" (じめじめ, jimejime). What grows in dark (i.e. gloomy) and damp places? Mushrooms, of course.
The expression from Haruhi there about telling Tamaki to "stop growing mushrooms in one's closet" which means, more or less "stop being depressed".