Yes. I have switched between plates and just bindings, on the same ski. I have done this on three different pairs of skis over the years. If you go from a free flexing plate to a straight binding, it will make the ski stiffer. However, going from a stiff plate like a deflex or old school EPB to a straight binding will make the ski softer. Also, a turntable heel will give you a softer tail by a little bit.
Yes, all these things are noticeable.
Many of my skis have just bolt through lifters on them and I have no problem carving them though.