I agree for the most part with the above post. However, I think the article on the provided link goes a bit overboard. It is definitely a bad idea to vent a bathroom fan into the attic. However, flexible ducting is commonly used for bathroom fans (not for dryers which should have smooth metal duct). It is also fairly common to terminate the duct by butting it up to a vent opening in the wall or roof. I have seen many installations like this and have never seen them cause mold or wood rot (the amount of humid air that would get back into the attic is minute). While it probably wouldn't hurt to insulate the duct, I've rarely seen that done, with no adverse effects.

Good insulation is good news. My point about the "beam" is to make sure it's in good condition and not cracked where you'll attach the electrical box. I had a similar installation in my kitchen, where the previous owner put up a 1x6 across the rafters and hung a fan from it. Apparently, they didn't drill any pilot holes before attaching the box with wood screws. I found that the 1x6 was cracked around 2 of the 4 screws, so only 2 screws were holding up this ~15 pound fan and light assembly! As for identifying the 2 wires, you can use an extension cord and a neon voltage tester. Plug the extension cord into a nearby receptacle. Separate the 2 wires at the fan. Turn on the power. Insert one test lead in the neutral slot (the wider slot) on the extension cord, and touch the other lead to each of the wires. The one that lights up the tester is the ungrounded (hot) and the other is the neutral (white.) Shut off the power when done. NOTE: This test assumes that the extension cord is plugged into a receptacle that is wired correctly. You can verify that with an inexpensive plug-in tester, available at any hardware store.

You should be able to remove he light kit and either and replace it with a plain cap or another light kit. We have done that in the past and changed the multi-lights to a single dome. If you simply need to have a way to suspend a shade, you might be able to drill a hole in the center of that cap and thread a piece to fit the shade to.

I wonder if the removal of the wall is contributing to the saggy ceiling. In our kitchen the ceiling was uneven, so the contractor left about an inch of space between the crown molding and the ceiling. He basically made his own ceiling line. In other words, ignore it. Old houses happen. In our previous home, the floors were so uneven that new guests commented and children giggled. The contractor put in a new, perfectly level kitchen. The kickplates were all jigged to match the floor. They looked like a fun house, but the counters were level.