Hi okibirdnerd,
I don’t know the specifics of the fabric they are using or the type of “padding” (I would guess you mean quilting) so Jim would probably be better able to answer these types of “feel” questions for you. If the fabric is a stretch knit then it would have little effect but quilting layers can have a fairly wide range of effects on “feel” so it will depend on the specifics. There is some general information about quilted covers and how they can affect the “feel” of a mattress in post #12 here and the posts it links to.
We had a dog named Sailor and he was often included in the “list” as well 
Phoenix