Yes, right-clicking while having the Stamp Brush will capture the tile (or a dragged rectangular area of tiles) from the map, so you can paint with it as well as see it selected in the Tilesets view. Since Tiled 1.4, this will also change the currently selected tileset when all tiles are from another tileset.
I'm not sure what you mean. If you're having trouble downloading Tiled from itch.io, try https://github.com/bjorn/tiled/releases.