It’s a combination of all three AFAIK. New engines of the same type built by the same manufacturer, engines that have been rebuilt and refurbished, and new designs put on older airframes to comply with new regulations for things such as noise.
Engines definitely wear out faster than airframes and they’re more susceptible to damage from things like bird strikes and FOD.
Yeah, it can get very complicated. Most of the time, engines for large commercial transports are "CFE" (customer-furnished equipment, as in Boeing's customer - the airlines). That is, the airline buys an airplane from Boeing, but then buys the engines directly from GE, Pratt & Whitney, CFM, Snecma, whoever. Like buying a fishing boat from one dealer and your outboard from another dealer. The airlines have tended to prefer that arrangement over the years, because then they can compete the engine suppliers against each other for every unique purchase (spares, parts, replacements, etc), rather than having a one-size-fits-all deal with Boeing as the middleman.
However, the FAA issues a "type certificate" (the license needed to fly an aircraft in US airspace) to an entire aircraft, including engines. So when the aircraft is initially designed, Boeing would certify the initial design with as many engine options as are "demanded" by the airlines, so that would be Kepler's R&D side - the airplane + engine is a package deal. Maintaining the engines and replacing them as they wear out (with new copies of those same designs) is then up to the airlines (the O&M side).
Airplanes do get "re-engined" all the time to extend life, meet new environmental/noise restrictions, better fuel economy, etc. (re-engining is industry jargon that definitely refers a new engine design, not just a 1-for-1 replacement with a new engine of the same type). Theoretically, a single airline could commission a new engine design and apply to the FAA for an "STC" (supplemental type certificate) to certify that new airframe+engine combination, but that's generally too expensive for a single airline to take on, and they don't have the same expertise (no laughing!) that Boeing does in working with the FAA. So in practice, what happens is that a group of airlines would go to Boeing and say, "we'd really like a better engine," and if there is enough market demand (i.e. we promise to buy XX more airplanes if you get this STC for us), Boeing would take on the task of obtaining the STC, which would then allow the airlines to buy new airplanes with that new engine installed (still CFE) and also install that new engine on their older airplanes.
Having said all that, the engine suppliers are incredibly incestuous, too, and there are lots of joint ventures, partnerships, etc, out there, not to mention licensing agreements to manufacture each others' designs. You may very well buy a GE design that has been re-badged as P&W or vice versa, but underneath it all, it has to be an engine of a "type" that has been certified by the FAA for use with that airframe.
Edit: military side is a little different, since in that case the "customer" (the military) owns the airplane design, the engine design, and everything else. On most programs, the military engines are "GFE" (government furnished equipment), same as CFE is for commercial, but since the military themselves is ultimately responsible for the safety of the design, they can go commission a new engine any time they like. The military is currently going through one of the most hotly contested re-engining programs in a long while, for B-52s. There are only 76 of them, but they have 8 engines each, so it's a big prize for the engine companies:
US Air Force launches contest to replace the B-52 bomber’s engine (defensenews.com)
