Okay.... I was under the impression according to what Solas says that he isn't a god. He says that perhaps the elven gods were some other beings that have yet been encountered but he makes the point that they weren't gods.
Also, perhaps his leaving without providing answers will actually place all of the inquisition in more danger. I don't see Lavellan giving up on looking for him. What if they accidentally interfere with his plans? I really think by not telling her the truth he is potentially endangering her more. I think what he did wasn't to protect Lavellan but to protect his plan. Weekes said in an interview that Solas stops kissing her at that crucial moment because he is about to lose himself and his cause. So he chose his cause instead of her and instead of telling her the truth like he had intended to do at that point, he makes something else up and then protects himself.
I think he is a god, but in the same way someone with a lot of power may be 'deified' by a large group of people...like the Inquisitor. So the term is meaningless to him because he knows how easily it can be manipulated and stretched. We'll likely get more answers as time goes on though.
Regarding leaving - I don't see the Inquisition looking for him beyond what Leliana has done -- popular headcanons aside. I imagine it will be much like DA:O where you carry on with your duty as Inqusitor and you will run into him or have to find him because of some otherworldly badness. He doesn't pose a threat at this point and didn't leave with anything crucial(EXCEPT LAVELLANS HEART), so to expend resources on a personal matter..'meh'. But I do understand the headcanons.
Why he didn't leave/tell her...there is a lot to justify that. His truth is MASSIVE. He probably has never been in anything close to this situation. How do you even begin to start telling them? How will they react? As Weekes said, he chickens out. He's scared of their reaction, of what they might agree to etc.
As you say, though, it may be blow up in his face in the end and he'll be kicking himself...again...but I think there's enough there to justify his decision at the time.