The weight changes throughout, so this means aerodynamics efficiency changes to maintain constant speed and altitude. Is this how most commercial jets cruise?
The short answer is yes, although altitude changes may be sought for various tactical reasons during a trip. The main thing that changes from weight reduction, from the pilot's perspective, is pitch attitude and thrust.
You usually file a cruise Mach number or TAS in your IFR flight plan (or at least the company operations dept does) and you fly that speed, more or less. If under radar coverage, you have some discretion to go faster or slower, but generally you will fly at the flight planned speed (you have to notify ATC for changes of more than 5% of TAS or .01 Mach).
If outside of radar coverage (far north, or oceanic for example), it's critical to fly the speed in your clearance within the tolerance because it's critical to maintaining separation. But in any case, in general you have the speed bug on the tape set to some target speed or Mach number and you maintain that with thrust.
When in cruise, the autopilot is in altitude hold mode, so it just pitches the plane as required to maintain whatever flight level it's set to. It doesn't care about anything else at that point. So altitude is held constant by the autopilot, and what happens is as fuel is burned off, pitch attitude will slowly go lower as less lift is required of the wings, pretty much imperceptibly, and speed will creep up with a constant thrust setting.
You're monitoring speed the whole time, and you will adjust thrust downward as required as speed slowly creeps up as your weight goes down, to keep the speed at the bug setting you've set (some airliners with autothrottle systems may do this part for you, as well as maintaining altitude).
Sometimes you'll fly through vertical air movement at altitude, like mountain wave, and if the wave you're in is going up, you'll observe the plane pitch over slightly to keep from climbing, and the speed creep up, and you have to reduce thrust accordingly to maintain bug speed. If you run into down wave later, the opposite happens and you have to increase thrust. Eventually you are out of the wave and back to the settings you started with.
Large commercial jets almost always fly IFR and thus are assigned altitudes they must hold so altitude is always constant.
There are no speed limits that practically apply to commercial aircraft at cruise altitudes (since there are no supersonic passenger planes left) so speed will vary a bit but many modern auto pilots are capable of speed holds so speed can be constant.
If a higher altitude is more efficient and the pilots have enough information to know that (winds aloft, local baro, etc) they can always request it from ATC but they may not simply climb to it on their own.
There was one famous exception to this rule. Due to how high it flew the Concorde was not assigned an altitude and was allowed to freely drift up as it burnt fuel and achieve maximum efficiency at all times. It could do this because there was no risk of it flying into another aircraft at the altitudes it cruised at.