Wiring in switch which turns on half outlet with no neutral
You evidently do not have a neutral in the switch box. The white wires attached to the switch are not neutrals but are either line hots or "switched hots" (the latter become hot when the switch is closed). The whites attached to the switch should have been marked with black tape to signify that they are repurposed hots. You can tell if the white are line hot or switched not by whether they are switched or not.

The neutrals for the existing circuit must be at the loads and were not carried into the switch box. (Neutrals are NEVER connected to one side of a switch with a line hot on the other side because when the switch would be closed this would be a dead short.)

In your top pic there is a white and a black going up. Is this new cable you brought in to go to the fan?

If you don't have a neutral in the switch box, then you cannot power a ceiling fan from wires in the box because you must have a neutral as well as a hot to power it. (And the neutral and the hot must be in the same cable.)

What you would have to do is bring in another cable with neutral and hot in the attic into the ceiling box for the fan. The neutral would be connected directly to the neutral lead of the fan. The hot would be connected to one of the wires in the cable you pulled to the switch box. This would give a hot in the box which would be properly paired with a neutral. The other wire in the cable in the switch box would be switched hot. In the ceiling box the switched hot would be connected to the hot fan lead.

EDIT The double switch you show could not be used to switch the fan by the method I describe because it has a common line hot connection. You would need a double switch that has two independent line hot connections.