Lucy, I have always considered these a weed tree. I don't find the lavender flowers particularly attractive and I think the seed capsules that hang on over the winter are positively ugly. I used to live in zone 6, where they would sometimes be winter killed, but the roots would survive. Last week I saw one at Wade & Gatton that had sprouted from a stump. W&G is zone 5 and probably close to the coldest area in Ohio.
Sooo... I think you should plant some. They're usually available cheap, they grow like weeds when young, and the young tree has huge leaves, which are kinda neat. If the things fail to die back over the winter, there is a technique called coppicing, or stooling, that you could use. See
http://www.rhs.org.uk/advice/profiles0105/Coppicing.asp Basically you cut the tree to the gound every year. So while the paulownia are providing some temporary shade, you can grow some real trees. Later you can paint the paulownia stumps with brush killer.
Sorry you have to go through this. Good luck.