SVS's downfiring subs have a "bass plate" that they 'stand' on.
Mine is on a berber rug and it just kills. And because of WAF and room layout and set up I likely have it in the worst or 2nd worst possible place. Hoping to fix that soon.
No, it doesn't matter. You'd need carpet many feet thick to start having any impact on bass frequencies. Any standard carpet will be "invisible" to bass frequencies that will go right through it as if it weren't even there.
I have an Hsu VTF-39 (down-firing), and it actually says in the manual that it is designed to be placed on carpet. It further states that if you place it on a hard surface, a small piece of carpet should be below it. It seems to work well in my carpeted room...