How do VMAX3 failover when a director is fault in a multiple-engine system

Both directors in an engine have access to ALL disks attached to that engine. So in this example if Dir 1 goes offline, Dir 2 in the same engine will take over all of Dir 1's disks. As you mentioned cache is mirrored to another director in another engine, so that data is also still accessible.

For example, Dir1 is mirrored with Dir3 normally. Do you mean that host IOs to Dir1 will be sent to Dir3 and then Dir3 transmits them to Dir2 after Dir1 goes offline?

The Front-end (host access) is totally unrelated to the back-end (disk access). Regardless of which director the host request comes in on, that I/O will traverse the virtual matrix to get the data of the correct disks regardless of which director those disks are attached to. So in the event that Dir 1 goes down, all data on all disks is still accessible via the virtual matrix.