You should not be able to touch your toes to the ground while seated. When pedaling, at the lowest point in your pedal stroke, there should be a very slight bend to your knee.

Where your saddle is relative to your handlebars is more a reflection of how your handlebars are set up, and not your saddle height. If your bars are too high, you can usually lower the stem. If the stem is as far down as it can go, then you can get a stem with an angle and just angle it down. FWIW, handlebars are usually lower than the saddle. The lower you can comfortably go, the more aerodynamic you will be (and it could actually alleviate some back pain, like in my case).

I highly recommend a fitting. Many a shop will do a mini fit for $50 or less (sometimes free) to adjust saddle height and fore/aft, handlebar height, and cleat position. If you bought your bike from a shop, try going back to them, if not, call around.