If I were to do this, should I use the PVC still or just buy a pump hose and snake it around to a y fitting on the washing machine line?[/quote]
Use the pvc and use the normal 1/4" per foot downward slope.
As far as the gravity drain possibility: After looking at your cleanout again it looks like your plumber used a sanitary T on it's back for your main stack. Here we would have to use a y but I'll assume that your area is ok with a sanitary T. If your main is 4" PVC then you could get a PVC fitting that is threaded to fit the cleanout like this:
Take out the cleanout cap and thread the fitting in place. Then glue on this 4x4x2 fitting
with the 2" opening either straight up or you could get by rotating the 2" opening 45 degrees from vertical leaning toward the washer drain if you need to minimize the height. (you'll need a short piece of 4" to glue the two female ends together or you might be able to find a 4x4x4 sanitary T with a street end like on page 12 of this pdf http://www.spearsmfg.com/prod_dimens...8_0308_web.pdf
Glue one of these into the 4" end and screw your cleanout cap back on.
Run the 2" line around the wall to a p trap on the sink tailpiece using appropriate fittings and keeping a 1/4" per foot slope towards the sewer line. (This line can serve both your washing machine and your sink so you don't have 2 parallel lines.) You can T up a 2" vent pipe by the sink and run it up above the sink level and over to hook into the vent pipe on your washer.
Another option would be to put one of these
on the vertical stack just above the existing cleanout but not as high as the drain from your washer. This might end up too high but is worth checking as the sanitary T on the vertical pipe will definitely meet code.
And another option is to carefully remove the existing 4x4x4 sanitary T with cleanout and replace it with a 4x4x4x2 sanitary T with right side inlet as shown on page 13:
It can be difficult to do this but it is possible. This option will put your 2" drain line the lowest of all the options at the sewer.
Most places do not allow a sanitary T to be used on a horizontal as yours currently is on your main stack. In most places they are only used on verticals and a Y must be used horizontally. You might want to check to be sure that your codes allow it's use horizontally or you will just be adding to the nonconformance and when it comes time to sell your house you might get hit with expenses to bring it back to code.
Codes vary from place to place so I highly recommend discussing this with your building department before glueing anything.