Thinking about what both you and Weem suggested might end up being the Creeper in this game. What he suggested, and what you added had me now picturing a scene-of-horror with how at random you would hear digging in dirt, in stone, and anything 'natural' and not playermade. How they would get spooked while panicking at what they might first assume to be a ghost. Maybe if you want to get spooky you could hear its 'war-cry' to send chills down your spine. If you want a time trigger then how about a dusk & dawn scare? The player, or players all panicking to keep their settlements safe by using their sword, bows & arrows, and such to keep it at bay. Even digging down to it to kite it around while it behaves all seriously scary and eerieness. A mixture of both the Creeper & Enderman in both spookieness and silence (at most times) that would catch you off guard. I can now imagine people building a strong basement flooring to keep their builds safe
Once Ages are added it shall be fun seeing low-Aged players struggling to deal with this monster during the dusk & dawn period aiming for them and their homes. Trying to dig away at the stone and dirt. Shall be fun trying to program the thing to lose interest and to not just go absolutely wild, to not try and tear down player-made blocks. Maybe it's higher tier'd buddy could do that? Maybe not seeing as how people would whine their creations are being destroyed. It should try and hunt you down, tunnel, or walk towards you while knocking down raw dirt and stone that you placed up during the brief dusk & dawn period to keep you on alert as you either wake up or ready yourself to go to sleep. Waking up to it standing in front in bed as you both stare one another down before killing one another in a Western stand-off.
Also, that mention of owls is something I'd love to hear at night for both visual and auditory vibrancy. Hearing and seeing it adding to the nightly vibe.