VC games are locked to the Wii on which they were originally downloaded to. You can copy them to an SD card but all this does is create a temporary copy on the card, you don't actually play the game from the card. The main game data is always on your Wii and the card just acts as a storage facility to free-up space on the Wii. I'm afraid you'll either have to get them to bring the original console to your house or buy them again yourself on your specific Wii.
As for how much space on an SD card the games take up, this is a little harder to answer. All the games come in various sizes, some may be smaller or bigger than others. The size that is usually referred to is block size, but this is specific to the Wii's internal memory. I wouldn't worry too much about it as a 2GB card should be able to fit quite a few games on it. You can always just stick a couple of games on at first and then put the card into a card reader on your PC and see how much space is being taken up but I'm guessing it won't be very much. Like you said, you could always get a bigger one as they are relatively inexpensive (I just bought an 8GB SDHC card for £11 on the net). You can get SDHC cards up to 32GB on a Wii, provided your system menu's firmware is version 4.0 or later.
Here is a lionk to a great Wikipedia entry on all things VC-related:
Virtual Console - Wikipedia, the free encyclopedia
It has a table further down the bottom showing which controllers you can use for which games (Snes, Nes, Mega Drive etc). You should be fine with Gamecube controllers but then again they may prefer the classic controller's shape/layout.