I'm not an EXPERT on this subject, I don't own a GC mod myself yet but am in the investigation stages. I have been a part of the scene for a loooong time, since the days when we didn't have modchips. All we had was the PSO exploit (and you could still get your hands on a copy of PSO I & II).
The 2 main contenders in the GC mod scene are Viper Team and Qoob, both having multiple chips (Viper GC/Viper GC Extreme, Qoob SX/Qoob Pro). There are some other chips that don't get much headlines, and I guess aren't as popular as Viper and Qoob.
Viper GC was the first GameCube mod chip. It doesn't have much flash memory to store homebew apps (such as loaders and bioses) and it has to be programmed via a serial port.
Qoob Pro came-out after Viper GC and improved on the Viper design by including a much larger Flash ROM size and the ability to program the chip via USB. I think you may be able to also stream games over USB with Qoob Pro as well, but I'm not sure. For quite some time, Qoob Pro has been far superior to the original Viper GC chip, just because of the added features. I have read though that it actually just uses a hacked Viper GC BIOS which I find pretty funny.
Just in the past few weeks (and not to be outdone by Qoob), Viper Team has released a NEW version of the Viper GC chip, called Viper GC Extreme. It too has a 16MB Flash ROM size, and a USB port for programming AND streaming (this I have read for sure). The new Viper GC Extreme, in my opinion after reading about each chip, is going to be top dog for quite a while. If you're going to base your purchase only on my advice, get the Viper GC Extreme.
To answer the other question, regarding the size of the discs required, it is possible with the current BIOS'es to use either size discs, the regular size DVD +/- Rs or the 3" mini DVD-Rs. If you want to use the full-size discs though you'll either have to cut your GameCube case or buy a replacement case that has been molded to hold the full-size discs.
I don't plan on modding the case, I've just placed an order for a buttload of Ritek Mini DVD-R discs (the brand that everyone says works best), so my case will still be stock.
Hope this info helps at least a little...