I don't know if PS has an automatic tool to split the image, but it has the crop tool which can be used much in the same way (crop-export; crop another part-export etc).
I believe that if you resize first and only after export to .jpg (or whatever) you should obtain a better result than if you first export to .jpg and then resize.
Also, after resizing, instead of "save as" try using "save for web and devices" (or whatever it is in English), that for some reason seems to produce smaller images even at the same quality. You may get a message saying that the image is very large and there may be slowdowns or something, but it should still work.