go to your local dmv, fill out proper forms and pay to get duplicates. In California you can pick up lost title (pink slip-DMV form is also pink for the lost title) In California you don't pay a fee...
Yes, but you won't get much for it - even if it's been restored to look like new you'll still only get around 40-50% of the value of a comparable car with a clean title. Probably better to sell it...
If it's not driveable it's worth 20-25% of the value of a comparable clean titled car. If it's reconstructed and roadworthy it's worth 60% of the value of a comparable clean titled car.
Some companies will let you buy liability insurance. almost none will offer full coverage. A salvage title means the car has already been totalled before.