How does one determine if an airplane meets airworthiness directives?
Compliance with Airworthiness Directives (ADs) involves an appropriate logbook entry. Examine the aircraft logbooks, or have someone competent in the subject matter—an aircraft mechanic, for example—to determine if applicable ADs have been complied with.

Each AD requires different compliance measures, ranging from a one-time inspection or parts replacement to recurring inspections. In each case, a logbook entry should be made to document compliance. Apart from the logbook entry, the aircraft or components concerned can often be inspected to confirm compliance, such as checking for a marked part ot that a placard installed. The details of what such an inspection could confirm would vary depending on what the AD requires.

In some cases, such as AD 68-17-04, which deals with Cessna stall warning systems, the AD calls for a one-time inspection followed by installation of a placard. The placard calls for the pilot to test the stall-warning system as part of pre-flight. In such a case, compliance with the AD extends beyond the actions of maintenance personnel, and will include pilot compliance with aircraft limitations—the placard.



Now, you seem to be asking three questions, beyond the title question of How does one determine if an airplane meets airworthiness directives? Let me paraphrase these questions:


How does one determine if ADs have been complied with?
How does one find ADs that apply to an aircraft?
How does one find "safety bulletins" for an aircraft?


The first question I have answered above.

The second question is primarily the purview of aircraft mechanics, especially IAs where applicable. The site you linked is a good way to find applicable ADs. It sounds like you may have run into some difficulty in getting the site to work correctly. A search for "Cessna 172" returns 21 ADs. Since ADs list aircraft or component applicability by make, model, and serial number range, among other criteria, each AD will need to be examined to determine if it is applicable to any given aircraft.

The third question is more difficult. Airframe, powerplant, and parts manufacturers may issue Service Bulletins and disseminate them at their choosing. Some manufacturers are easier to deal with in this regard than others. An exhaustive answer on this topic is beyond what is appropriate here, in part because it is constantly changing. If in doubt, contact the manufacturer for what you need.