Can I apply for the green card from here or do I have to go back to my country?

You can do it here as long as you have not accrued unlawful presence.

That entirely depends if you are in our out of status.

You should be able to apply for adjustment of status here in the US, but you will need to be sure everything is filed correctly and you have maintained your status.

It depends if you are in valid status in the US and will be at the time you will be eligible to submit the adjustment of status application on the basis of being selected in the DV lottery. Since that's not right now, you will need to see when you are eligible to file this and whether you will be in lawful status at that time. If you are not going to be lawfully present, you will not be eligible for AOS in the US and would have to go through consular processing.

Yes you can apply for your green card while you are in the U.S. You do not have to leave the country.

My suggestion is that you contact an attorney and discuss your options BEFORE you make any plans. In order to submit your application for adjustment of status in the US, you must at that time be validly maintaining your nonimmigrant status. The DV visa that you won cannot be available until October 1, 2012 at the earliest. Yes you can file a request for an extension of your tourist visa, but I would suggest that the filing of such a request would be inconsistent with the claim that you want to remain in the US for more time as a tourist. Tourists leave, DV visa winners stay as residents. So there are a number of very important concerns that you must be aware of before deciding what to do. My experience with DV cases is that it is always an enormous difficulty getting them to work while the DV lottery winner is here in the US. It is always one thing after another and a lot of this has to do with the fact that the DV system is set up as one where the applicants are overseas. USCIS is not set up to handle these cases as effectively as is the US Consular Office and this is part of the constant frustration that I have always encountered with these cases. I would highly recommend that you contact an attorney and that you make it someone who will explain everything to you so that you can make an informed decision on how to proceed. Don't guess on this one because you have a ticket in your hands that you want to use most effectively. If it is not done correctly, you may lose the opportunity to reside in the US that has been provided to you by being picked to receive a visa. Good luck.

Yes. I've done these types of cases before. As long as you are still in legal unexpired status, you can apply for adjustment of status (residency) immediately without having to return to your home country.