Am I correct in understanding that the higher the altitude the thinner the air? If so, does this have an impact on the ability of the aircraft to remain "held" in the air? I guess what I am asking is does the higher altitude impact aerodynamics because of thinner air?
An aircraft is a series of juggling acts, with various factors competing for control.
Basically, an aircraft flies because of INDICATED air speed (IAS). That's the actual effect of the air on the wing. What you'd feel if you were able to stick your hand out the window. If the best airspeed is 250 KIAS, then it will (to a degree) be the same at all altitudes...
But, the air is, as you say, a lot thinner at altitude, so to achieve that 250 KIAS I will have to go faster through the air mass as I climb. At FL400, it would be about twice as fast. So, that gives us TAS, or true airspeed. For a constant IAS, the TAS will increase as you climb, and for a constant TAS, the IAS would be reducing.
The aircraft only cares about IAS...that's what makes it fly. TAS is what gets us from point A to B (once we correct if for the wind, which gives us GS, groundspeed).
Overriding all of that is mach number. That's our speed as a percentage of the speed of sound. It gives one set of maximum speeds. It is related to temperature, and basically reduces as you climb. It's TAS.. At sea level it will be around 680 knots, reducing to about 560 at FL400. So, that 560 kts of mach 1 is roughly 280 KIAS, which is not much faster than our airliner's best IAS. Drag rises dramatically around mach one, so it is limiting for a number of reasons. Above about .85, you'll slow to below the best IAS to keep away from it. In a climb, at about FL300 the climb IAS will hit mach .8 (or so) and from that point you climb at a mach number.
An IAS, at a given weight, will give a fixed 1G angle of attack. An airframe has a 'best' angle of attack, so we use IAS to indirectly give us that best AoA. As an aircraft burns fuel, and the weight reduces, it needs a lower angle of attack to produce the required lift. To maintain that best 'alpha', the speed is reduced...which explains why cruise speed is constantly being reduced as a flight progresses.
But.....the engines have an RPM at which they function best too (actually close to full power, call it 90%). So, reducing the speed means the engines may not be at that best power setting any more, so after a while you'll be better off climbing. The 'best' altitude would be the one at which the IAS giving that best AoA simultaneously requires the engines to be set at their best RPM. The upshot of this is that you end up with a series of climbs and decelerations during a flight.
Hopefully a little clearer than mud.