Can I run a 60A sub-panel off of another 200A sub-panel?
Yes, you will want to come off of the panel inside the house. The one on the outside is likely just a shutoff. In which case it would be unsafe for you to open because you cannot dissconnect the power at the meter. So at least one side of the breaker would be hot. Flip the main cuffot outside so that you can work on the panel inside without worry. 

For a 100' run I'd use service cable. 6/4 should be fine for 60 amps. You will need to run four wires. Two will carry the two phases, one will be a neutral, and the other will be a dedicated ground between the two panels. You may also want to add a ground at the new panel. Ground rod is best, or tie it to a copper cold water line (old school).

As for the appropriate breaker for the compressor, that will entirely depend on the compressor. If it runs on 240V then yes you will need a "double pole" breaker so that you grab 120V from each phase for a total of 240. But you will need to check the motor on the compressor to see if it requires 240v or 120V. It should also say how many amps it will pull. 

Yes you could add a double 30 breaker and additional circuits for tools lights or whatever. Really you can have as many circuits as you like as long as the total used at any one time isn't more than 60 amps. So for example you could add a 30 amp circuit for a compressor and a 50 amp circuit for a welder and you will be fine as long as they don't run at the same time.

Edit: wanted to add that for the 60 amps panel you will need to use a 60 amps double pole breaker in the main panel to tie into.