It rather depends on your definitions. Although he was involved in some dangerous activities with his pupils the ultimate responsibility for the safety of lessons has to rest with the headmaster, similarly he didn't decide on the Forbidden Forest detention, he just carried it out and went to some lengths to make it as safe as possible in the circumstances.
For sure he does have a bit of a 'blind spot', as the book put it, for assessing the potential danger posed by magical creatures, but it's not as if Hogwarts is that safe a place anyway. Quidditch is plainly potentially lethal. Indeed his lessons have a pretty good safety record compared to Hogwarts in general. Equally, you can't really call this 'unstable' as his skewed judgement is, if nothing else, consistent.
Equally, there is no indication that Hagrid doesn't care if his students are hurt and he seems perfectly able to form reasonable social relationships with his colleagues and pupils. Consider that when Draco was mauled by Buckbeak (through his own arrogance) his first reaction was to take him to the hospital wing.
He also shows a lot of empathy for other people and animals, which is actually pretty impressive when you consider how he has been treated for most of his life.
He was also able to bounce back pretty well from being wrongly imprisoned in Azkaban and that in itself suggests that he is reasonably mentally resilient.
It is also reasonable to say that a certain amount of fearlessness is pretty much essential for his job and indeed the whole order of the Phoenix.
Indeed the worst act of unthinking recklessness in the whole book is Dumbledore putting on a cursed ring, is he mentally ill ?