I would think you should be able to do all these things with the HCH, and it will be much easier to find fitted items to do this because it's much the same as the standard Civic. I want to say I remember reading the spring rates for the hybrid are slightly different for the added weight of the batteries, but I wouldn't think the size differs too much. Please note the Civic is nothing like the Insight as far as handling. It's a lot softer and the car is larger. Granted the handling is way WAY better than the 04 Prius I drove (no horrible body roll), but if you wanted a handling car the Insight is way better. The Civic is what it is, a family sedan. You could still autoX it, but it probably wouldn't be as fun.
As far as the stereo, ouch.... it looks very heavily integrated however there is one huge difference in the Civic's stereo, it does not suck! You get 6 speakers and a decent amount of power and it sounds like it was made in 2006 and not 1906 like the Insight's stock stereo sounded. From someone who had a shoebox sub in his Insight and the whole 9 yards I'm happy with the stereo in the Civic, and I love my music.
To the nitty gritty, the head unit is their own custom box in the dash, it's by no means a standard size. There are also the steering wheel mounted controls that integrate in to it. You also get an aux input. If you get Navigation though it does look to be a standard double din size unit, but again you get even more buttons on the steering wheel for the voice interface. There may be some hope if you are set on swapping out the head unit. If you look on the regular Civic Sedan's page they have some pictures that look to be the double din opening with a standard Din size deck and a 6 disk changer in the dash:
http://automobiles.honda.com/models/aud ... ivic+Sedan
The only problem is the Hybrids interior colors are specific to the hybrid so these trim pieces might not quite work... but there may be some way to order it with a double din opening on the dash short of getting the pricey navigation option.
Reguardless it *could* be done:
http://civictour.honda.com/vehicle/index.aspx