I found Sega Genesis / Mega Drive 101: A beginners guide on Racketboy a great resource.
Is the Genesis exactly the same as the Mega Drive?
The name is just a trademark issue, it was originally the Mega Drive, but was marketed as Genesis in North America since the name had already been registered.
What's the difference between the 1, 2, and 3 versions?
Model 1 - Only version to include a volume control slide switch for the stereo sound output. Supposedly, this model is the easiest to perform modifications on for regional bypasses, overclocking, LED change and S-Video output.
Model 2 - Smaller, simplified design. Stereo output is included in the A/V outputs and there is no volume control for the sound.
Model 3 - Even smaller, harder to hack, not compatible with Sega CD or 32X.
There's a forum thread discussing which of the three is the best model. There's even more information on the different Mega Drive systems on Wikipedia:
During its lifespan, the Sega Mega Drive quite possibly received more officially licensed variations than any other console. While only one major design revision of the console was created during its lifespan, each region has its own peculiarities and unique items, while other variations were exercises in reducing costs (such as the removal of the little-used 9-pin EXT. port) or expanding the capabilities of the Mega Drive.
Variations of the Sega Mega Drive
The article covers all of these variations.
Can I buy a Genesis, and then buy cartridges either for the Mega Drive or the Genesis?
The games are locked per region, but there's a good point on this:
System modifications such as adding region switches and SCART outputs can be readily accomplished on most model 1 and model 2 hardware.
Coincidentally I was considering buying a Model 2, which is also the one with the most votes on the mentioned forum thread, so I guess this pretty much answers my question.