What happens if a private Cessna 172 pilot tries to land at big airports like JFK or O'Hair when there is no emergency?

yes, but you would be very restricted.

The type rating is required in aircraft that need additional training. It is a safety of flight issue not a for hire issue.

Older planes are usually very good purchases, IF they have been well maintained over the years. Airplanes have to be inspected annually by an A&P mechanic with an Inspection Authorization so they are usually very well maintained. Any purchase should include having an A&P mechanic review the airframe and engine logs and evaluation of the plane, including looking at how many hours the engine has since overhaul and checking compression. I had a 1966 Cherokee 140 for several years and finally sold it when I bought my current Experimental aircraft (a BD-4). It helps a LOT if you can get an A&P license and maintain you own plane though.

It is very common for small airports not to have a control tower, radar, or communications equipment. There are very well developed procedures for "uncontrolled airports". There is a standard traffic pattern that aircraft fly at almost all airports (consisting of a downwind, base, final, and upwind leg) and there are specific radio calls that are supposed to be made at certain points in the pattern. Most airports have a fixed base operator to supply fuel and services. They often monitor the common traffic frequency and supply some info to pilots about wind direction and runway in use. There's also a specific way to enter the pattern - usually at a 45 degree of the downwind leg, Yes it's possible for aircraft to collide and it happens several times a year - usually when a low wing airplane is above a high wing airplane in the pattern. Neither can see the other so occasionally that can happen.

Technically is you are flying domestic you are not flying over foreign airspace. If flying IFR you must file one. Needless to say when filing a flight plan you must decalre if you are domestic of international flight.

You're stuck with a water landing (ditching). Hopefully at altitude the pilot can see a ship and make the landing in the vicinity of the ship for quick rescue. It so rare for a plane to lose power to all engines though that you should never hear of such an incident. Depending on the route, however, Bermuda may be within gliding distance.

Using the old (pre GPS) method you would use the sectional and plotter (aviation ruler) and measure the distance. Using a GPS you can usually turn on distance rings on the screen or if you have the airport selected as destination you can read it right off the screen. If it's your home base you should learn the landmarks and their distances from the airport. In the Miami area, there are so many airports and landmarks, if you're flying there, during your preflight preps you should measure out distances to some landmarks that you plan to pass over and mark them on the chart or flight log.

I suspect that the idea of max climb plus glide is not a new one, and that, if it really were cost-effective on fuel, we would know about it.