From your description you have a number of serious code violations that are resulting in the problem you describe. Fortunately it will be very easy and fairly inexpensive to change it.
With the singular exception of a standpipe such as a laundry standpipe, no vertical input to a trap may exceed 12" in lenght. (Yours is currently 14".)
S-traps are prohibited.
Under the International Residential Code "IRC", the maximum developed lenght of an 1-1/2" horizontal line from the trap weir (point where water spills out of a trap) to the vented stack or horizontal waste line may not exceed the diameter of the pipe divided by the required pitch. The required pitch for an 1-1/2" line is 1/4" per foot so the maximum lenght is 1.5" / .25" = 6ft.
Under the Uniform Plumbing Code "UPC" that maximum length is derated to 3'6". (Currently your maximum developed length from the trap weir to the stack is in excess of 15'. This is the reason our line is not venting correctly.
The codes require a cleanout at each change of direction greater than 45deg. Currently you have a 90deg bend with no cleanout.
The Uniform Plumbing Code will allow a maximum of one (1) air inlet valve (commonly called a studor vent or cheater vent) per structure. Under the IRC once a structure has the required Main Vent you may use an unlimited number of AAV's.
Before we begin discussing how to resolve your problem allow me to explain how PVC fittings are identified.
PVC piping may be used for both water supply piping and DWV (Drain, Waste & Vent) applications however pressure and DWV fittings are not interchangable.
All PVC pressure fittings have sharp bends where they change direction and they are identified by the angle of the bend. I.E. 90deg elbow, 60deg elbow, 45 deg elbow, etc. (These may not be used on a drain line.)
All PVC DWV fittings will have a slight radius in the direction of flow or as in the case of a WYE fitting the pipe enters at an angle directing the flow. In order to help identify DWV fittings versus pressure fittings, DWV fittings are identified by a "Fraction of a circle" thus a 90 deg elbow for DWV piping would properly be called a "1/4 bend" (360deg / 4 = 90deg thus a 1/4 bend)
Most pipe fittings have a female hub on each end to accept a pipe input, but some fittings have a female hub on one end and "male spigot end" on the other end. This permits connecting one fitting directly into another without the need of a section of pipe between them.
Now let us start on your project:
Begin by cutting the 1-1/2" horizontal pipe below the floor and remove the 90deg elbow, veritical riser and S-trap and tailpiece up to the disposal.
You will need more horizontal clearance at the disposal to install a P-trap so I would suggest you begin by attaching the P-trap to the disposal outlet and the new trap adapter on the P-trap. You will then be able to measure the required horizontal clearance to determine where your new vertical riser from below will need to be.
Drill a new hole through the floor for the new riser.
Now go in the basement and extend the existing 1-1/2" horizontal line to a point where you can attach an 1-1/2" Wye with the side inlet on top of the pipe. The inlet will be at a 45deg angle to the pipe so you will now install a Street 1/8th bend (45 deg) into the Wye side opening and adjust the 1/8th bend to give you a vertical opening under your hole. Install a section of 1-1/2" pipe to make a new riser up into the cabinet. On the open end of the wye you will install a Cleanout thread adapter and a cleanout cap to close the line. You may use a "Combo" in place of the Wye & 1/8th bend. A combo is a Wye & 1/8th bend made in one fitting to simplyfy the job but either method is equally acceptable. Care though, code prohibits installing a "Sanitary tee" on a horizontal line.
When you have the section below the floor completed you may need to install additional hangers to support the pipe. You may use either the preformed PVC J-hooks or you may use vinyl perf strapping but you may not use any wire or metal hangers to support plastic pipe.
Moving up to the sink now, measure the height were a "Sanitary TEE" will fit with the side opening pointing towards the trap. (You may not use a Wye &18th bend or combo on a vertical riser).
You may attach the trap adapter directly into the side inlet of the Sanitary Tee or you may extend the horizontal line over towards the P-trap as needed, providing you do not exceed 3'6" horizontal from the tee to the trap. (The IRC would allow 6 ft but at 3'6" you will be good for whichever code is in your area.)
From the top of the Tee extend the line a minimum of 4" vertical and install a cleanout thread adapter on the top of the pipe. You can then screw a studor vent into the threaded adapter to provide your vent and since it is easily removable it meets the requirement for a cleanout also.
NOTE: I prepared an illustration but the program will not let me upload it. Send an email to
LazyPup@yahoo.com and i will reply the illustration. In the mean time I will contact Troy and see if we can get the illustration posted here.