The city tore up our entire road and put in new sewer lines and told me that our house shared the same lines with the neighbors and that we don't have a clean out and that we had to pay to have someone come out and put new lines in for our house. My laundry room floor keeps flooding with sewer water from all the houses and I am wondering who is responsible for fixing this because our house is not up to code and we were never informed of this?

I will assume you own the home.

It is your responsibility to investigate your home when you buy it. In older areas it was common to have joint sewers with several homes built years ago.

So it will be your cost to put in a new sewer line assuming yours go onto others property. Now if yours go into the street and the other peoples are connected to yours.
So how are they ran, who has the actual connection on their property

In most municipalities, the home owner is responsible for everything from the home to the street connection. Check your deed and or the property site plan to see where the shared pipes are. Good Luck.

City provides you with sewer lateral which is capped about 6' past your property line. You have to connect your sewer into this lateral. Call plumber to connect it for you. In the meantime, you should not be using any plumbing fixtures in your house or it will back up into your bathrooms.