How does a full fuselage and structure inspection happen in practice ? (for modern Civil airliners)
The FAA "Class D" maintenance/inspection of commercial airliners, which must occur every 6 years, is extremely thorough:


The cabin trim is completely removed to inspect the airliner's skin and superstructure from the inside. Seats, safety equipment and control system linkages are also inspected in detail with each individual cable or wire tested.
The plane's exterior paint is removed to inspect the exterior skin, including rivets and seams.
All critical moving parts on the exterior of the plane (engines, gear, ailerons, flaps, spoilers, rudder, elevators, shock booms, etc), are removed from the plane, disassembled and visually inspected for wear or damage, then re-assembled and tested.
All pressurized lines (hydraulic, water, fuel) are removed from the plane and each part individually visually inspected before being re-assembled and pressure-tested in the plane.
The cockpit is stripped, all moving parts in the control column and throttle assemblies visually inspected, and all flight instruments visually inspected and tested.


There are many more maintenance/inspection tasks than I list, I'm just going through the major ones. The plane is more or less stripped to its bare metal frame, every component visually inspected and tested, then the aircraft is reassembled with individual systems tested again on the airframe. The process can take up to two months for each aircraft, and can cost between 5 and 7 million US dollars. Most major airlines have either their own in-house maintenance crews or the manufacturer perform this maintenance, with a team of FAA-employed inspectors overseeing and signing off on all the work. 

Most airliners receive 3 Class D inspections in their lifetime; at the fourth, occurring around 24 years of airframe age, the cost of a Class D is typically not considered worth it given the aircraft's remaining value, and the aircraft is usually retired and either sold to smaller non-US airlines operating in jurisdictions with less stringent standards, mothballed and stored for potential replacement parts value to the rest of the fleet, or sold for scrap. Some airframes in service with some airlines have been kept through four Class Ds in order for the airline to have sufficient time to procure replacements; American's remaining MD-80s are largely newer airframes acquired in the TWA merger, but there are a few of American's own deliveries left in the fleet which are past 24 years and are still around mainly to allow American sufficient time to work through the bankruptcy, United merger and procurement of replacements (Boeing's 737 production lines are going full-tilt and American is not the highest-priority customer, so American had to split its order in this class between the 737 and Airbus's A319 and A321).

Because of the level of detail and deconstruction required, including complete cabin trim and paint removal, airlines often use this maintenance interval to perform any desired upgrades or tweaks to their fleet configuration. Common upgrades in recent years have included:


adding winglets for vortex reduction (reducing drag and allowing closer spacing at busy airports),
upgrading avionics and even control systems (some older airframes can be upgraded from direct cable-driven hydraulics to electro-hydraulic controls allowing partial automation of flight with EFDs), 
upgrades to cabin trim including re-spacing of passenger seats (more or less legroom or "slant", balancing between class seating), 
replacing galley equipment and other secondary systems that may be a different standard from the rest of the fleet, 
adding new passenger conveniences (such as satellite-based Wi-Fi, headrest IFE, etc) and 
new exterior paint schemes for corporate rebranding (such as American's ongoing rebadging from their decades-old AA silver to the new American white, and Southwest's recent move to the "Heart" scheme from its older "Canyon Blue" and the original red/gold before that). 


Aircraft acquired in airline mergers are often the highest priority as the airline seeks to rebadge and standardize the acquired fleet to operate alongside its existing planes (and ensure no corners were cut by a struggling airline that would create liabilities going forward).