First of all, I don’t think “Luke sneaking into his apprentice’s room” is a huge deal to start with. It’s his nephew, it’s not that creepy. Not to mention it wasn’t his intent to kill Ben when he entered, I honestly doubt he even considered the possibility beforehand.
Second of all, it’s not like he hacked into his laptop and looked at Ben’s search history. Luke was searching Ben’s feelings, this is something Jedi do practically 24/7. He just decided to do it when Ben was asleep because that would give him the clearest picture. Is that a bit of a breach of privacy? Sure, of course it is (though not to the extent you’re making it out to be). Luke was afraid of Ben’s potential. He wanted to find out what was in his heart, and what his future was. Could he have handled the situation better? Yes, but that’s the point. In his hubris he thought he could take matters into his own hands. When he realized his mistake, it left him with shame and consequence.
Yes, but for me the problem is, that a Jedi should be able to control his or her fear, and make rational decisions, especially one who has seen and experienced far worse than one student’s dark feelings. Luke is acting out of character without having been provided the proper motivation. He is a Jedi Master afterall, not some novice. I can accept Luke becoming emotionally compromised (like Mace Windu was in ROTS), but considering Luke’s history and experiences, it would have to be something pretty shocking, like Ben willfully injuring one of his fellow students, and seeming to enjoy the experience. If Ben had stormed off after doing something that terrible, I can picture a shaken Luke sneaking into Ben’s room at night, still angry about what has happened, wanting to protect his students. The anger would cause Luke to ignite his lightsaber, once again showing that the dark side is always a danger even to a Jedi Master. That would have worked for me, as Luke was then properly provoked, then tempted, and subsequently controlled his anger, even of it was too late.