The "Gremlins" novelization sheds even more light on Gizmo's sanity. The backstory of the Mogwai was that they were genetically engineered pets created by aliens (yes, aliens). Why they just didn't stick with a Furby is a mystery for the ages. The Mogwai's asexual budding when exposed to dihydrogen monoxide was simply a feature to reduce production costs. As a side note, this implies a non-capitalist civilization, since it makes "bootlegs" of the pets extremely easy to produce, as shown in the film. This is, of course, assuming the Mogwai come from a environment that is not scarce on water. You never can tell what those pesky aliens are like.
However, it seems the genengineers weren't completely successful, since the Mogwai had a couple of flaws, namely their sensitivity to light and their changing into "Gremlin" form when fed after midnight. The most drastic letdown was that the "cute, cuddly" personality they were going for turned out to be fairly rare (only about one in one thousand).
So, since as an alien Gizmo cannot fairly be judged by human psychological standards, we must compare him against his own kind. His behavior pattern clearly falls into the minority, though probably not to the degree of a full-blown psychosis. Based on his Rambo transformation in Gremlins 2, it seems the potential exists within Gizmo to fit in with his kind, although this only manifested under extreme circumstances.
Therefore, based on information from the "Gremlins" universe, we can only conclude (sadly) that Gizmo must be considered insane. However, the good news is that this can be resolved by simply spawning off a large number of Mogwai from him, and putting them all in a nice quiet place where Billy or Kate (and especially Mr. Futterman) can't get to him. Maybe provide a car battery and some jumper cables just to speed things up. I'm sure his loving offspring will do everything they can to "cure" Gizmo.
Source: "Gremlins" novelization by George Gipe (c)1984
memory errors likely.