The award is related to the costs incurred.
This varies from each case and each individual. Know that going to trial can take years, but many times in return the judgment in court is larger than in a settlement.
No. Tell the insurance company that you will get back to them. In the meantime, contact an attorney immediately. Often an insurance company will offer a minimal amount of money in return for your signature stating that you will not sue them. Never take an insurance check without first consulting an attorney.
Normally, in cases such as these, you do not pay the attorney any money until, and if, you win your case. If you do win, they are likely to request 33% of the settlement.