Hi there,
Firstly local speed of sound decreases with altitude due temperature decrease with altitude. Equation reads;
LSS = sq.root(Absolute Temp)
I'm sure there may be other factors that affect LSS but for simplicity we just consider the absolute temperature as the variable.
To the second point, why does TAS increase with a constant IAS as you climb. Well the difference between IAS and TAS is a density relationship. As you climb the air becomes less dense due to the decrease in pressure. Hence the aircraft's actual speed becomes greater. Remember density is proportional to pressure and inversely proportional to temperature but the decrease in pressure is the overriding factor.
Final point, why does IAS decrease with increasing altitude? This is the case when you start climbing at a constant Mach number. You changeover to climbing at a constant Mach number at something typically like 25000ft due to reaching MMO sooner than VMO as you correctly stated. Now if you climb above 25000 temperature still decreases. Now go to the following equation;
TAS = Mach No. x LSS
Mach No. is constant, LSS decreases with altitude due temperature decrease and therefore TAS decreases. If TAS decreases do must IAS because pressure is still decreasing as you climb. This is still the case as you pass the tropopause.
Hope that makes sense!