Problem #1 is that you have an S-trap, which is no longer permitted by code. To meet the current requirements, the trap will need to be vented. Proper remodel would be to install a drain and vent stack into the wall framing so that a proper horizontal trap arm can be installed into a sanitary tee in the wall.
If you are not doing a remodel, you should find out if your plumbing code will allow the use of an air admittance valve. Unless you are installing a vanity, you are still going to have the pipes showing under the sink.