A* is just like Dijkstra, the only difference is that A* tries to look for a better path by using a heuristic function which gives priority to nodes that are supposed to be better than others while Dijkstra's just explore all possible paths.
Its optimality depends on the heuristic function used, so yes it can return a non optimal result because of this and at the same time better the heuristic for your specific layout, and better will be the results (and possibly the speed).
It is meant to be faster than Dijkstra even if it requires more memory and more operations per node since it explores a lot less nodes and the gain is good in any case.
Precomputing the paths could be the only way if you need realtime results and the graph is quite large, but usually you wish to pathfind the route less frequently (I'm assuming you want to calculate it often).
