Legal Question in Immigration Law in New York

Re: F-1 to Green Card through Diversity Lottery

Yes, you can either apply for an immigrant visa in the US consulate in your home country (consular process), or apply for adjustment of status (if you don't plan to leave the USA until you have it approved).

Becoming a USA lawful permanent resident should not affect your studied in the college. You will no longer need a F-1 visa, instead you will have a green card.

Re: F-1 to Green Card through Diversity Lottery

I would need more information regarding your particular case before I can comment on it properly.

However, your F-1 student status should not be affected if you continue taking your classes for your visa.

With regards to leaving the country that will be a problem.

The normal procedure is while your immigration papers are being processed by the Immigration Service you must seek their permission (they will give you a travel document) in order to leave the country. There is no guarantee, however, that you will be let back in this country our border.