I'll answer your questions first, then ask you a couple.
1. Yes, you can find double switches for a single gang box, but the result is you have a whole lot of wire crowded into a very small space. This is hard to work with and not a lot of fun for the wire either. To pull out your existing jbox, a sawzall is wonderfull for this. Grab one from a friend if you can. figure out which side of the box is mounted on the stud, then cut out drywall for the 2-gang box to give yourself some wiggle room. Then, slide the sawzall blade up into the bay between the box and the stud and cut the nails. The box should swing away free of the stud, assuming the drywaller didn't mud the box to the drywall. If he did, use your utility knife to cut the mud out so it will pull away. Now, after you cut the nail wings off of it, you can insert a replacement PLASTIC 2 gang box and just screw it to the stud through the inside of the box. Voila, no drywall repairs!
2. Rather than ripping out an existing wire only to replace it with 14/3, just run a new additional 14/2 alongside the original run. It will be easier to wire this because you don't have to undo what's already there.
3. As far as wiring,you should have a source cable coming into the jbox that had the black wire going to the switch, and the white wire nutted together with the white wire coming from the ceiling light. The black wire from the ceiling light should have gone back to the switch.
Now what you are going to have is the source black wire nutted to (2) 10 inch black pigtails that each go to a different switch. The black wire from each of the ceiling fan cables should go to the other side of each switch. All of the whites are still wire nutted together.
4. Instead of having to use the pull chain to adjust the speed of the fan, you could install a dimmer switch rated for your load (you will have to ask a qualified electrical parts salesman what that is) instead of a regular switch, and you could install a dimmer for the lights as well. (Just a suggestion!)
Keep in mind, I am not a licensed electrician, and as far as I know, none of the other regulars on the board are either. This information is presented in good faith, but you use it at your own risk. You may choose to consult other sources before proceeding. Contacting the fan manufacturer is probably a good idea.