Can two neutrals on different legs of the same circuit be connected together to shared neutral of ceiling fan?
There is a simple rule of thumb that keeps you honest, about cabling: No loops.  What you just drew there is a loop.  You can't have loops. 

Think about a tree - trees have many places where they branch or split. But you never see two branches re-merge so they are connected and interchange life fluids.  That doesn't happen. 

So all wiring must be a tree - unlimited splits involved, but no re-looping back.  

This setup has a loopback.  It must be cut somewhere to be legal. 

Mind you, the cut can be at a device or switch box, by having either a physical or imaginary "partition" separating all non-ground wires). Safety ground wires are allowed to cross the partition/loop/web, but never neutral. 

So if you can obtain a fan with separated neutrals for fan and light, or if the fan's instructions include a procedure for having the same effect,  then you can indeed have the lamp+its neutral powered off the 3-ways and their accompanying neutral, and the fan+its neutral powered off the branch coming over from the lamps. 

The problems with looping are serious -- buzzing wires which then causes metal fatigue, cracking and overheating or arcing...  heating in metal bits near this wire, and high EMF which can trouble animals or people who are EMF-sensitive. 

When the "tree" plan is followed properly, all currents in each wire are balanced by an equal and opposite current in the other wire(s) of the cable. This causes the EMFs to cancel each other out, because that's the way magnetism works.