but Mach needs also on temperature
M depends only on CAS and pressure height. No OAT input required. Plenty of older threads with the relevant equations quoted if you so desire.
mach is needed to display SAT
stagnation temperature rise depends on M. So M is an input to convert TAT to SAT (with whatever recovery coefficient is relevant to the installation). Likewise plenty of threads with the equations
real A/C will have a tendency to pitch continuously in cruise with A/P off,
you are probably thinking of the phugoid. Bit of a roller coaster motion which is only a minor nuisance to the pilot. With the A/P height lock in the phugoid degenerates to a similar period speed oscillation unless the A/T takes care of it. Some aircraft have a pronounced phugoid, others are quite benign.