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The best comedy bit I've ever seen in a show (from a personal perspective) is in fact, Mr. Lancer swearing in classical literature titles. It's always so perfect for the situation and makes me cackle every time. Whoever thought of that deserves an award.
And for Danny he could be
1: Suspiciously eyeing Jason. Why is this ghost/halfa/liminal dating his teacher? Is it a plot to get to him? Spectra tried being an employee, maybe the new plan to get close and hurt people is to date an employee. Well not on his watch! This 'Jason' guy is going to be so closely watched, Danny will learn his every secret. Every. Single. One. No one hurts his class + teacher!
2: Mr. Lancer is dating a ghost/liminal/halfa? Mr. Lancer protects them all from the anti-ecto acts? Does that mean Mr. Lancer and Jason could accept him? Jason just pulled a gun on a GIW agent. Yeah, could you sign right here? No, this isn't an adoption form, what are you talking about.
Either Jason coaxes Danny in like a feral cat, or Danny 'Batman Adoption Bait' Fenton is doing his level best to get adopted, despite Jason's confusion (and deep desire to never let his siblings know).
Jason doesn’t want to bring Lancer anywhere near his nightlife and Lancer is perfectly fine with that. Lancer manages to convince Jason into some martial arts and combat training to defend his students if need be. It’s pretty concerning how often he has to use said training.