Why is the compressed air (from engine compressors) taken from High Pressure (HP) stage instead of Low Pressure (LP) stage?
Coming from the compressor section of the engine, the bleed air is sent to the precooler. Even air from the "low" pressure section needs to be cooled before it is sent to the air conditioning system. Here is a precooler from a CitationJet:



The page with that image notes that the bleed air is "hot enough to melt insulation and alter the heat treatment on aluminum alloys." The precooler reduces the temperature of this air and regulates the pressure to the pack system.

When the engine is at low power settings, bleed air is taken from the high pressure section. The engine must run fast enough that this provides enough pressure for the bleed system to operate properly.

Once the engine gets to a higher power setting, the bleed air is taken from the low pressure section. At higher engine speeds, this air still has the pressure required to run the bleed system.

Since the bleed system needs a minimum pressure to operate, compressing the ambient air to this pressure will always add a certain amount of heat.

Here is a diagram of the bleed system, to help visualize what is going on. Note that even after the air goes through the precooler, it is still hot enough to be used for wing anti-ice.