Don't they mean "put it on the table"?
It doesn't make sense in English without a direct object, but I get the impression that it is often OK to leave it out in Swahili.
No. Nyumbani is not a verb so you cannot simply put verbal prefixes on it. I like your idea though. It reminds me of Turkish evdeyim (ev-de-yim = house-at-I'm).
Nina nyumba. = I have a house.
Nina viazi nyumbani. = I have potatoes at home.
Niko nyumbani. = I am at home.
To say where something is, you at the subject prefix (except that a- changes to yu-) to the syllable -pa for exact location, -ko for inexact location or -mo for internal location. -ko tends to be the most common I think.