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havent fitted my HKS coilovers yet, but i have to question your coilover experience (curiosity killed the cat :) )
Installing a coilover / lowering your suspension does not "make" your gear changes smoother or makes your braking responsive.
this is purely due to the vehicle center of gravity (due to lowering and stiffer spring rates and damper settings) - Therefore when braking - this will actually increase the time /distance for your car to actually stop - due to the forced weight of the car being lowered (and obviously speed) which causes extra weight to the front wheels.
The steering will feel heavier (again due to the lowered vehicle) but you will gain better handling/ feel when cornering - BUT you may also experience extra sway - depending on how hard/soft your dampering is and other suspension parts or setup (RSB, endlinks camber, strut braces etc)
the rule of thumb is when you lower your vehicle - you should always take into consideration other suspension parts that will suit your driving style and feel safer on the roads, circuit etc.
Brake upgrades - Rotors, Pads, calipers, brake lines etc
Other suspension parts - F/RSB's, bushings, endlinks, camber kits, strut/pillar braces etc.
Tyres - wet/dry grip conditions?
I found that the stiffer suspension removed the "lunging" of the car on gear shifts, it sits flatter and firmer, so the shifts seem to lose less momentum.