Wii games can be played from USB on both Wii and Wii U (in vWii mode).
There are N64 emulators out there, you'll have to check with them to see if you can store the ROMs on USB or if they have to be stored on SD. I've not used mine in some time, so I don't know off the top of my head.
Gamecube, currently cannot be played on vWii since all known methods of USB loading GameCube games require a GameCube controller. So this currently can only be done with a Wii using one of two methods: Dios Mios and Devolution. There are guides on the site on how to use both. Both require a FAT32 with 32k cluster partition to be the first, primary, active partition on the drive.
No need for any hard modding. Everything you want to do can be done with soft modding. We have guides for Wii (Softmod Any Wii) and vWii. Links to both are in my signature.
If xbox 360 and PS3 allows for FAT32 hard drives, they should be able to live together. If they requires some odd-ball formatting, then no. Wii wouldn't be able to read it. Old Wii guides have you make a WBFS partition on the hard drive, which would not be read by anything else. This practice is no longer recommended. I recommend FAT32 all the way, others may say NTFS as it gives some pluses, but has a significant number of minuses are well (guides for both FAT32 and NTFS also exist in my signature).
Playing online is not affected on the Wii. I can't speak for the vWii, but I don't think that it would be affected either. Maybe someone else who has done this can chime in.