1. how long I can stay in US with out status
You should depart the US immediately if the H1B is denied!
2. If I appeal or file the petition, will I be able to continue staying here?
Yes, but it is very risky. Let's say the appeal is denied after a year, you will be out-of-status for 1 year and you may be barred to enter the USA for up to 3 Years!
3. What is he possibility of getting through this case
Please post a copy or text of the NOID, I will provide more information, I know you probably have 30 days to respond!
4. What other alternatives that I should look for to stay in the US until the case is adjudicated?
The best solution is to immeditaely transfer the H1B to another Employer - You may send the resume at email@example.com (I can help, please use H1Bsupport as the subject line)