You should break up floor and run pipe in wall and vent it and stub out for sink. But if you go to floor you need a S-trap vented if you have one unvented if you dont. It is against the code, if you go to sell your house you will have to get it done right.
As RP says, S-traps are illegal in new work. Many jurisdictions allow them to be “grandfathered” in as old existing work done when such traps were approved.
As you undoubtedly realize, there is insufficient room for the usual P-trap fittings to be installed under a pedestal sink that drains into the floor. One solution would involve cutting the P-trap fittings to try to fit them in while still retaining the P-trap configuration. Sometimes you can install a “Redi-Vent” at the top of the waste tee, if your local authority allows it, and you may also be allowed to use those rubber drain fittings, if they remain exposed under the sink.
In all cases, have the final job inspected and approved.