I'm assuming a non-turbo engine since at sea level with the turbo not working the intake temps will be 'normal'. But when at altitude when the turbo is working, those hot gasses will definitely be affecting the temp!
You didn't use the equation properly. n is equal to the moles of the gas, not density. They're not the same thing at all. You basically calculated the relationship between temperature, density and pressure of the standard atmosphere. If you didn't round to -5 degrees your answer would be -4.8 degrees Celsius, which is precisely the temp at 10,000' for a standard atmosphere. Temp and pressure both affect density and that's all you showed.
Manifold pressure, under 'perfect' conditions, would have a maximum of 29.92" at SL. When the airplane climbs up, that maximum pressure would equal the outside atmospheric pressure. At 10,000' it would be 20.58". If we are setting 65% power with 2400RPM lets say MP has to be at 20" at SL. At 10,000', based on what I've seen in performance charts, the required MP for 65% power at 2400RPM, would be lower than 20". The only reason I can come up with for that difference is that as you climb in altitude, the pressure ratio decreases faster than the density ratio. Meaning, the density doesn't decrease directly proportional to pressure because of the lower temp at altitude, which would negate part of the pressure drop. Meaning, you don't need as high a pressure to get a certain density into the cylinders since the lower temp increases density.
I can see why one might think that lower back pressure would require less MP but I haven't seen any proof of it. It makes sense but I could only see it having a VERY small effect.