All the above is about right. I will add that above Mne you will encounter mach buffet and then a high speed stall as the shockwave over the wing starts to destroy lift. The thicker part of the wing will go first. This is invaribly the closest part of the wing to the fuselage and on swept wing aicraft will cause the nose to drop, causing mach tuck.
The normal way to operate is to climb the aircraft at a fixed IAS, typicaly around 280 to 300 kts and then into a mach number typicaly .72 to .78. depending on type and weight.