Dpi depends on how the map will eventually be viewed. The standard for high quality printed maps is 300 dpi. That assumes the viewer will be looking at the image at less than arm's length. For a poster-sized printed map, 150 dpi might be enough, although it depends on the size of the details.
If the map will never be printed, dpi doesn't matter at all. In that case, only the actual pixel dimensions matter, and they should be based, again, on how you want the map to be viewed. If you want the entire thing to fit on an average screen so that the viewer doesn't have to scroll, you should shoot for maximum dimensions of 1024 x 768. That's really quite small, though, and if you're shooting for that size, I'd suggest you actually design it at least twice as big and scale down right before exporting your final image (don't forget to undo back to full size before saving your master document, though!)
The resolution of individual elements is going to depend on what size you ultimately choose to work at. Size the elements appropriate for the image you are making. Some people like to design their brushes and stamps at twice their working resolution so that they can increase the size of the canvas they use later on and not have to redesign their brushes. The only hard and fast rule is don't scale up. You'll almost always lose quality if you increase the size of a raster image. So it's almost always best to design things at a larger size than you'll need because you don't lose much quality when you scale down.