The horizontal run in the floor must be at least 2", but cannot be larger that the pipe which it is connecting to. You can clean out the hub in the existing CI, or you can snap it off and use a fernco coupler to make the connection.
The washer standpipe and sink drain in the picture will meet code pretty much anywhere.
You do not have to go up with the vent right there at the laundry. The vent can run horizontally (above the flood rim and with pitch) to a location that will make it easier to go up through the roof.
Venting out the side wall is not allowed anywhere that I have been. Fumes from the sewer main will be coming out the vent, thus the reason it should be up through the roof.