Existing Ceiling Fan fixture doesn't work - 2 hot wires, 1 neutral all confusion
Yeah, I see what they did now.

First off, they are feeding the right switch from a junction box in the ceiling on the white wire. Then the black wire on the right switch is a switch leg back to the closet, linen, receptacle. So, this all works because the neutral for that is in the junction box in the closet, linen, or receptacle. Wherever it is fed from.

Then they jumpered the hot white wire to the left switch. So, when the left switch is thrown it only energizes the black wire and there is no neutral to the light box. The white wire has been cut off.

The problem here is there is no neutral in the switch box to power the ceiling fan. 

The fact that they used three way switches to do this is not a problem since they wired them as single pole switches. They could be easily replaced if you wish but it will not fix your problem.

You would need to get a cable from the feed junction box to the switches that has 3 wires and a ground. So, you would have a hot feed to the both switches, a switched leg to the closet,linen,rec., AND a neutral for the left switch to power the ceiling fan. There are other ways to do this that may be easier but the concept is the same. You need a neutral wire.

Good luck and stay safe!