Many people recommend only getting an attorney involved once you've been denied; many attorneys won't touch a case until it's been denied, actually.
Attorneys typically can't shave off much (if any) time, unfortunately.
Re the payment terms: Their fees are capped by law, and they typically get paid before you do once you're finally approved.
For Social Security Disability, the attorney receives 25% of the claimants back pay with a cap at $6,000. If the claimant is not approved, the attorney is not paid. But, the attorney can charge for things like copies of medical records, postage, copies, etc and that is not covered by the Fee Agreement.