Should my foot be able to step flat on the ground, when I sit on a bike seat?
For a standard bike in normal use you should not, from the seat, be able to touch the ground (without leaning, or except, perhaps, on extreme tip-toe).

A standard diamond frame (with horizontal top bar), for road use, should be sized so that you can stand flat-footed over the top bar with a "comfortable" margin (but no more) between the bar and the stuff above the bar.  (A bike used off-road will generally be a bit more compact, and, of course, the top bar is often not horizontal.)

The seat should be positioned so that your leg is almost fully extended at the bottom of the stroke (the knee should not go perfectly straight).  

One way to fit the seat height is to have someone hold the bike while you pedal backwards with your heels on the pedals.  With your heels on the pedals the knees should go perfectly straight at the bottom of the stroke, but with no need to rock side-to-side to maintain contact with the pedals.  (Once you switch to using your toes on the pedals in a normal fashion the proper almost-straight leg extension is obtained.)

Off-road, and for stunt riding, generally the seat will be a bit lower, and some may find it convenient to lower the seat a bit for city riding.  But be aware that a too-low seat puts significant strain on the knees and can lead to knee injury (especially when combined with riding in a too-difficult gear).