1. No GameCube controller support, obviously. That's about the only thing that's different, the games still function the same. It works like GameCube mode works on Wii: It is essentially a Wii, no Wii U controllers or features will function and you'll need Wii controllers.
2. If you ever want to upgrade your VC games to the Wii U versions when they release, then you'll need to do the transfer. Now that homebrew works in Wii mode it isn't a reason to keep stuff on a real Wii anymore.
3. Yes, LostWinds will remain on your old Wii, as reported by people who have it. You can try searching GameFAQs to see if anyone had trouble transferring the unlisted games, but I just looked and I didn't see anyone who had such a problem. When you do the transfer you will see the list of all your games that are being transferred, so you'll be sure that it's moving over.
4. It doesn't matter what's on your internal memory or SD card, you're not transferring the actual games. You're just transferring the licnences to the Wii U, and so you will have to re-download anything you want to play. The Wii mode has the same amount of space as the Wii, so you won't be able to store more games than a Wii can.