"Matěj is expecting a child with František's daughter."
Translation:Matěj čeká dítě s Františkovou dcerou.
Here it is literally doing something with someone together, not using him/her as an instrument. (Notice the case is named instrumental!)
The with was assumed in "to do something with something" / "dělat něco něčím" but that is a completely different meaning of with. It is an instrument, not a companion. If it is a companion you have "to do something with someone" / "dělat něco s někým"