Game Design is not an umbrella term for jobs in the game industry.
There are different sorts of disciplines in the games industry. Art, Design, Music, Programming, Marketing, Production... I've no doubt there are more. Read more about what sort of jobs there are, and what is required in them at FAQ #7.
So you don't want to learn game design, you want to learn game art, specifically 3D (at least, from what I gathered). Knowing this may make it easier to search for information relative to the path you're focused on.
Well, that depends. They certainly never do any harm. Doesn't take too long to build a solid foundation for you to work with, if you're willing. Do you want good drawing skills? And what are you looking to model? Environments, characters? Guns, armor or other gear? Crates?
1.Do I need good drawing skills ?
A free sculpting program and a free modelling program.
2.With what software should I start?
If you're a student, grab some of Autodesk's Student Version programs, 3Ds Max and Maya are both good choices. But again, it depends on what you want to model, how you want to go about it... Sort of need more information.
There are good tutorials for pretty much every 3D software I can think of. If you're thinking 3D, remember that a lot of tutorials can be translated to whatever program you use, unless they use a specific feature you don't have.
3.Are there tutorials for that software?
Yes, there... There are a lot. I cannot suggest any, though, as I mostly use reading material.
4.Are there videos to explain/help you about the business.
Here are some links to places I go most frequently to find said reading material (aside from the oft linked Sloperama FAQs):
It's midnight here, so I might have left out a thing or two, but hopefully there's something of worth in here somewhere. Good luck, mate.