No that's a really good question. Weight and balance calculations for flying are only concerned with the location and value of the variable weights - fuel, pax, baggage. The C of G of the "empty" aircraft with all those components is considered a single unchanging value for the purpose of establishing a starting point before loading, and is established just by weighing the airplane while empty. The weight of the components is a factor in where the empty C of G will end up, but once established the entire "empty" machine is treated as a whole unit for W&B purposes.
Vertical C of G is not normally provided with A/C weight and balance data. Although the location is not too big deal with most types of aircraft, for gyrocopters it's quite important because they need to have the engine's thrust line as close as possible to the center of mass for safety reasons (if you unload the rotor on a gyro by pushing over or from turbulence, and the thrust line is significantly above the vertical C of G, the thrust line tries to spin the mass of the machine like a pinwheel about the C of G and the gyro tumbles, with fatal results; the tall and spindly look you see on small gyros today is driven by the need to mount the engine/prop lower to get the thrust line close to the vertical C of G).
On a gyro, the procedure is to hang the loaded ready-to-fly-state machine by the rotor hub, establish a vertical reference line below the hang point, then hang it by another point at the same level that is offset forward or aft using a special "2 hang point" rotor hub fixture, or alternatively if it'll work you can tip the machine back on its main gear until it balances, and establish a vertical reference line extending up from the gear contact point. Where the line extending down from the rotor hub, and the line extending up from the gear balance point (or the two downward lines if you had a dual hang point fixture) intersect, is the vertical C of G.
To figure this out on a 172 you would need to do something pretty radical, like support it along the fuselage at one point using a suitable resting fixture just forward of the main gear but aft of the C of G, and tip it back until it balances and establish a vertical reference through the balance point, then move the fuselage rest forward to ahead of the C of G and tip the plane nose down to establish a new balance point and vertical reference. Then where the vertical axis of each balance point intersects, there's your vertical C of G. Of course you'd want to do this while loaded with people, fuel and bags and you'd probably have to pay them or offer them free beer, after you've found an airplane to use, built a fixture, and all that. You could also probably do it by weighing on scales with the nose propped up high on a block so it points up at an angle, then reweighing with the mains propped up high on two blocks so it points down on an angle, then do some clever geometric calculations to work it out.
Or, if you are as lazy and cheap as me you could call Cessna's Tech Support organization and tell them what you are doing and what you need, and somebody somewhere should be able to find the data (hopefully) without charging you for it.