DAL is a Southwest hub. Southwest could swap a plane if the inbound to DAL is delayed.
But going along with the scenario as is:

As soon as your first segment (DAL-PAX) is delayed to the extent that you will misconnect, you will be automatically rebooked and be notified that you can choose to rebook yourself.
When you file the claim, you submit the original itinerary, the updated itinerary, and the common carrier delay statement.
Your delayed DAL-PHX flight will have a delay reason assigned to it and that common carrier delay statement need to state a reason that is one of the covered hazards.
What caused the delay to inbound to DAL is irrelevant to your claim. (That said, Southwest will most likely assign the same reason to the impacted downstream flights).

You can get a room in either DAL or PHX - depends on the new routing you picked to rebook into.
- If you rebook into a new itinerary that departs DAL the next day(s), you're fine to claim the hotel room in DAL
- If you rebook into a new itinerary that still departs on the same DAL-PHX flight, but not departing PHX until the next day, you're fine to claim the hotel room in PHX.