If you modify a report by adding filters, does it update a dashboard tile from that report?

No$,$ the tiles in a dashboard are only a snapshot of your data.

No$,$ you will have to remove the tile, refresh it$,$ and add it to the dashboard again.

Yes, it will update a tile, provided you create the tile using Pin a Live Page in the More menu.

Yes, it will update a tile by using either of the two methods to refresh.