Can I cut out a 12" section of a footing to run new plumbing in a basement?
You can do it, but you may want to check with your local building official for local requirements.
The normal way to do this is to core drill through the wall/footing and insert a protective sleeve two standard pipe-sizes bigger than the drain.  For a 3&quot; bathroom drain that calls for a 6&quot; nominal Sch40 pipe which has a 6-5/8&quot; outer diameter.  That's a big job and you may want to rent a core drilling rig for it or hire somebody to come do it.
You could also use a rotary hammer; in fact, you may need one anyway if you have to chip around rebar to cut it out.
Why is your sewer line not low enough to go underneath your footing?  That seems unlucky.  If the sewer line isn't very long it might be easier to replace it.  Try to find out how deep your city connection is and whether re-laying it would be an alternative before cutting through your footing!