Electrical – Low-voltage cabling through ceiling? Up to code

At first glance, I see several problems:

1. The junction should be in a junction box.
2. Is there even an electrical nut on the live wires? Hard to tell under all that electrical tape, but it doesn't look as if there's room for one.
3. Is the neutral from the red wire at top going into that tangle of lives? If it's carrying live, it should be tagged to indicate that.
4. The red wire to the right shouldn't be running along the top of the ceiling joist. If you can, run it along the middle of one of the sides.
5. Ideally, the yellow wire at bottom would pass through a hole drilled through the joist. Failing that, there should be furring strips either side to prevent you from stepping on it.
6. The electrical tape on the neutral and ground wires isn't strictly necessary for code, but if it's there, it's customary to have it go around the nut to help hold it in place.
7. Thanks to @tester101: different sheath colors indicates different gauge wires. Check that your circuit breaker has the appropriate amperage rating.
8. Staples: if you run wires across the joists, they should be stapled to keep them in place between the furring strips (that you also need; see #5). You'll also need them along the length of the wires that run along the joists, and it might be a good idea to have one near where the wire drops down through the 2x4 header (given the quality of the rest of the job, whoever did this may not have used staples when running the wire through the wall cavity).
9. It looks as if you might have a lot of slack in those wires. Wiring shouldn't be piano-wire taut, but you don't want enough slack that you can trip over it.

Just because the inspector didn't mention it, doesn't mean it's OK. They're human, so they can miss things, especially in an attic where the lighting isn't good.

If the ceiling is concrete, you can mount the fan to that. Fan boxes are meant to support the fan, but a concrete ceiling will support it just fine. As long as there's enough room in the box for the wiring, use the 4" fan bracket to mark two holes outside the box on the concrete that won't interfere with the wiring coming out of the box, and use appropriate tools and fasteners to attach the fan bracket to the concrete around the old box.

If you can't modify the box, chances are good, being embedded in concrete, that it'll still hold the heavier load of the fan designed for the bigger box, and you can modify the fan bracket, or use an adaptor.

If none of these solutions seem workable you can buy a 4" box, and mount it to the 3" box, then mount the fan to it. It's not ideal, as the fan mount would no longer be flush to the ceiling, but it should work just fine, and provide all the room you need for wiring.