1. You need to find out if your H-1 was applied w/ COS or not. If it was applied w/ COS and same gets approved, then you will have to start working on H-1 from COS approval date (typically Oct 1). If not, then you can continue working on L-1
2. Yes, you can ask them to withdraw the petition. As the filing fees is not returned in case of withdrawal, you will have to figure out how the companies reacts to the financial h i t.
3. Legally, companies are required to pay for the H-1 filing fees. They cannot demand those fees from you, but they can ask for attorney fees.
Another option is to let the H-1 get processed. If it gets denied, you don't need to worry. If it gets approved, then you can travel outside of US and return on stamped L-1 visa after Oct 1 to continue on L-1 visa. Your H-1 will become kind of dormant and you can go to H-1 in future.