The domain is obvious for t>0. The two roots to the equation are a small negative number which we need to discard because of the t>0 condition and the other one is 18.2668s, I assume the one you mentioned. So you are correct indeed on this.
For the second part whether this is a realistic model I would say that there are some strange things about the equation. Consider the height at lift-of, it is 0.9m. This is acceptable. Consider the time for which the height is maximum. This can be calculated by considering the derivative of the equation with regards to t equal to 0. This gives approximately tm=9.1332s and thus a height of h(tm)=hm=9177m. This seems not right, going more than 9km high in 9 seconds, I could be mistaken, I'm not a rocket expert.