Is there an easy way to count the number of installed apps in my Android phone? Maybe this info is in some file. I can also have it mounted in my computer and run a bunch of linux commands.
Was about to say the same as Firelord – he missed the "ease" part, by the way, and has you counting lines manually. Rather try adb shell pm list packages -3 |wc -l (when on Windows, better use quotes: adb shell "pm list packages -3 |wc -l"). Takes a few seconds, and then shows the number.

Example, with one of my test devices:

Linux $ adb shell "pm list packages -3 |wc -l"
15
Linux $