Don't think I can help with it all. One thing though...
While you ask about compressors, looking at the engine as a whole helps clarify matters.
Compressor section hits air with a big fan, giving it additional heat, kinetic energy etc. The total energy (you cite kinetic and static) is therefore not constant and thus a change in one energy form of the air does not necessitate a change in the other as it may be supplied externally (via the compressor blades).
To tie it all up,
energy is imparted to the air via the compressor,
which is powered by the turbine,
which in turn is powered by the combustion of fuel and, importantly, air,
which is compressed by the compressor.
A bit of a 'foot-bone connected to the knee bone' but in a big circle. The magic bit is getting all this efficient enough so that there's enough energy left in the air (after turbine has turned compressor) to get thrust.
Not directly what you asked, but it hope it helps. I think the bit you're stumbling over is keeping track of what stays constant and what doesn't, which is definately the b@st@rd of it all. If it's turning something, being turned by something or having fuel burned in it the the energy state of any one thing is likely to be changing and thus 'this goes up, so that goes down' arguments can be misleading.
Anyway, I'm talking way above my pay grade but until you get owt better, hope it helps