Extending plumbing vent horizontally
Consider an Air Admittance Valve, rather than a traditional vent, in this situation.  The downside is these are mechanical devices that in theory can fail.  The upside is reduced sewer venting to the atmosphere and less pipe.

Buy a quality one: there are $3 models that are not the same thing.  Look for a series of certifications for this task and expect to spend at least $30.



A vent that goes up and then down is not OK.  AAV's are approved in the International Plumbing Code, and the USA National Plumbing Code.