2/ Why does increasing the compression ratio (piston) or pressure ratio (turbine) improve efficiency?
...Carnot's Theorem tells us that in reality even with an idealised engine (no friction losses etc) efficiency is determined by the difference between the temperature at which heat enters the engine and the ambient temperature of the surrounding environment. Since we cannot really hope to change the ambient temperature of the environment the best we can do is improve the temperature at which the heat enters the engine. In aircraft this is done by increasing the temperature at which the fuel air mixture is ignited.
You will not find the answer to your question in the Carnot cycle where, for two given heat reservoirs, you can achieve the same efficiency regardless of compression ratio used. With your statement above you are effectively hoping to use heat from the hot reservoir to increase the temperature difference between the two reservoirs. A hopeless endeavour.
If you want to consider the effect of increased CR in the Otto cycle with a notional perfect constant volume combustion process occuring just after TDC then you are simply left with a longer expansion path, a correspondingly longer compression path and therefore the difference between the two is also maintained over a greater path ie its more; more net work.
But you cannot increase the theoretical efficiency of the Otto cycle by taking energy out the crank and feeding it back to increase the
temperature of the fuel/air charge prior to combustion. This is where slippery_pete has given you a bum steer. If you look beyond the cycle itself to the combustion process then the picture changes because you can increase the speed of combustion by increasing the temperature and pressure in the combustion chamber.