1. You didn't say if this was a week day or a weekend. We landed around 10:00 on a weekday and it took us about 35 minutes to get to Casco Viejo. We left Casco Viejo on Saturday at 5:00 a.m. and it took about 20 minutes to get to the airport.
2. You may also consider Ancon Hill. It has a beautiful 360 view from the top. While we were there we also took a side trip to Panama Viejo, which to me was just okay.
3. Can't help you there.
4. We used private transportation for the airport and it was $35 each way. I'm sure you can negotiate a better rate with a taxi. We did a city tour, which included transportation to Miraflores Locks and it was $50pp.
5. Well, it's not entirely a "local" eat but Casa Sucre coffee house is super good. Their lunchtime specials include a beer with any sandwich. It's about $10. We ate lunch there twice during our three day stay. We were also staying at their adjacent property.
I did see some other places where it looked like a number of locals were eating. I'm sure you could try one of those out. People seemed really friendly, especially if you are able to communicate in Spanish.