Unfortunately, in order to apply for an adjustment of status in the United States based on DV lottery, one must be in a lawful status to begin with. At the same time, if you leave the United States now, you will be subject to a three or ten year bar (depending on how much of unlawful presence you have accrued thus far). Your best bet at this point is to stick with your asylum application. In any case, speak to an immigration attorney for more detailed analysis of your situation.
No, there is no such thing, if fact by leaving the United States, you will set two things in motion:
1. Abandonment of your pending asylum case and,
2. Statutorily applicable bars for re entry since it will trigger once you leave the country.
Your F-1 will not be processed internally as you fell out of status to begin with. Also, you are not eligible for AOS via I-485 as there are no stated grounds in your discourse.
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