Workflows, Validation Rules and Before triggers fire on Lead Conversion only if the Setup > Customise > Leads > Settings > Enable validations and triggers is checked.
It might be worth checking if this setting is enabled, because otherwise the Lead Status value will not be synced to your custom field on Lead Conversion via workflow, and therefore not be copied across.
(Even then, I believe the Lead Triggers / Workflows fire after the Account and Contact ones, so it may be to late for the Lead Status to copy across)
The order of execution at Lead Conversion has been answered here
This can be achieved by a trivial LeadAfter trigger, or if the Lead Status is always set to Qualified upon Conversion, then you could set 'Qualified' as the Default Value on the Contact Lead Status field.
(One way of doing it also may be to only allow 'Qualified' Leads to be Converted - this can be controlled by having a different layout for Qualified Leads, when the status changes to Qualified, a workflow fires - changes RecordType to QualifiedLead and therefore associated Page Layout changes, which now has the Convert Button. With this approach, the Lead Status at conversion is available in advance and available to be copied across at Lead Conversion)
This appears to be related to the default record type. When logged in with a user that has the Record Type with 'Enterprise' process assigned as the default, I see 'Identified' as the conversion status.
There is a setting under Lead Settings: App Setup > Customize > Leads > Settings.
This setting says "Do not change lead status values to new owner's record type on lead conversion". Marking that TRUE will allow any user to set the conversion status for the lead process assigned to any record type.
Further, I found the metadata for Lead Settings is not supported, so you cannot deploy that setting from one environment to another.
Best Answer
I'd expect "Dead" to be the equivalent of a "Closed-Lost" Opportunity and would use the "Dead" value similarly as a means of closing the record for a Sales Path. You could also "hide" the value by using a record type in which you don't include the value of "Dead" for the Lead in your Sales Path.
If you do need to still include it, move the value in your picklist to below "Converted" so that it's in the equivalent position of where you'd locate "Closed-Lost" for an Opportunity. In that manner it will appear after "Converted" in your Sales Path and won't be one of the "stages" that your Sales Path must move through in order to reach "Converted".