You can save a workspace which will preserve the choice of index pattern, fields, colours, icons etc and also any vertices/connections currently loaded in the workspace. This is a "disconnected" view meaning that the graph may include a selection of vertices e.g. IP addresses that are no longer present in the index/indices behind the index pattern. Only when you try drill-down from these vertices will it become apparent that they no longer exist in your index. This is by design - items of interest such as rogue IPs may come and go in the underlying indices but remain an item of interest in saved workspaces.
If you open a saved workspace, the URL in the browser will change to the unique ID of that workspace and this is a URL that can be shared.
Not currently. The standard "ctrl-F" search facility of the browser can find and highlight text labels inside the SVG used to render the graph but lacks the pan/zoom required to bring them centre stage.
When you drill-down from Graph to other parts of Kibana you can generally preserve any visualization configuration choices in that visualization because they are held in the URL. The trick is to configure the target visualization e.g. Discover with the choice of settings then copy the URL from the browser's URL input and paste that into the Graph settings - see this demo.
Another quick fix for the Discover tab specifically might be to use an index-pattern for Graph or more general use where you specifically choose in the index settings: "no, I do not want to use the time filter".
Hope this helps.