Is it possible to, basically, fast-forward games on PCSX2 in order to quickly bypass parts of the game that otherwise cannot be skipped?
Nope. You can disable frame limiter (Tab or F4 keys in your keyboard) and if your PC/settings can handle the faster speeds then you get fastforward but there's no way to force it.
Ugh. A "Skip Cutscene" button really should be mandatory for video games. Thanks.