I don't drive a Honda so I can't say for sure how coilovers will ride.However,I run coilovers on my 91 Mustang GT and they don't bounce at all.A nice strut/spring is always gonna provide a better ride and handling but its not a necessity.Not sure if they are offered for Hondas or not,but KYB,s and Strange are great shocks if you're looking for something cheap that works good.You can buy shocks that are non adjustable(stock)single adjustable,or double adjustable.Compression and rebound are adjustable on double adjustable shocks.I bought a kit on ebay that included 2 aluminium threaded collars(tube with threads that slips over top of strut) Two large nuts that screw onto collar and sit underneath spring(these are what you turn to raise or lower the car)2 150lb rated springs,spacers,strut mounts,torrington bearings,and spanner wrench(used for turning the nuts to lower car) This is probably the same type of kit you're talking about.Yes you can install this kit over your existing shocks
I'm pretty sure it will work because we did that with my racecar. There's plenty of travel in the strut or shock.