..are they any houses that are built on a state border. As in, one half of the house is on Tennessee, and the other half is on Kentucky, or something like that?

Google it.

You can't build houses on top of parcel lines (pre-made land plots made to make property management easier for the government and to avoid this very situation).  Before you can build a house the government has to come and survey the land.  If the proposed location sits on top of a parcel/tax jurisdiction line, they won't grant the build permit and will most likely just ask if you'd relocate x meters that way.

If your house is built on top of a parcel line, your residency is dictated by the location of your bedroom.

What if someone lives really close to a timezone? and they live in one (lets say Central) but they work in another (Eastern) how much would that suck.