How to stabilize speakers without spikes.

Use Harmonix footers under the spikes. They are expensive but actually improve the sound (at least in my case it did, tightening bass and adding some needed warmth).

I'm with Warrenh for the most part, but I would say it depends on what you are coupling with. Hardwood floors (especially those without adequate subflooring) move around a lot so decoupling might be the best bet in your case. There's really only one way to find out for sure and that's to experiment!

As far as the acoustics being altered with the new floors- chin up! Some of the best studios in the world implement a little audio action with brick, stone, wood, and other reflective surfaces to liven up an otherwise dead room. It isn't exactly a reference attribute with regard to playback but it can often make for a more enjoyable system so long as it's not too out of control. A large rug and your furniture along with the wood floors might just end up being an improvement over carpet alone.

Good luck.

I would try Cerapucs by Finite Elemente. They do not have spikes to damage the floor and come in several sizes. See info:
http://www.musicdirect.com/products/detail.asp?sku=AFINCERAPUC

type in adona in the audiogon search and scroll down to his floor spikes and protectors for some ideas.

If you don't have a carpet anymore, there is no reason to have spikes since there is nothing for them to bite into and improve stability. Just use the brass discs alone or coins. Hockey pucks might be a better solution for a wood floor like someone else suggested.