The migration assistant tool would copy all the configuration data including the vDS to the new appliance. However, I would recommend having the config exported prior to the migration attempt. You would need to schedule a downtime for the migration but not separately for vDS.
Does it mean that during migration (including the dvs configuration), there will be no connectivity for the VMs?
The vCenter Server deployment utility will not automatically upgrade your DVS version.
DVS is configured from vCenter Server, but the data plane is on ESXi hosts. This happens by deploying a "proxy switch" on ESXi for each DVS you configure on vCenter Server.
So upgrading vCenter Server won't cause any interruption to your ESXi hosts or the running VMs.
After you successfully upgrade vCenter Server, you have the option of upgrading the DVS version to 6.0 or 6.5.
As per the following official document, it won't cause any disruption to your VMs:
Quote: "The upgrade of a distributed switch is a nondisruptive operation, that is, the hosts and virtual machines attached to the switch do not experience any downtime."
The link you posted belongs to 6.7, where it says that "The upgrade of a distributed switch causes the hosts and virtual machines attached to the switch to experience a brief downtime. For more information, see KB 52621."
This is only for 6.7, so since you're upgrading to 6.5, you shouldn't notice any network traffic disruption.