I was wondering, for a correctly sized mountain bike, or hybrid, should I be able to place my foot flat on the ground, when I sit on the 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).
Typically your feet should not be able to touch the ground flat when you're on the seat - it is a sign that your seat is too low (you may be able to touch the ground with your toes depending on your shoe size - I wear US 13 and this is possible for me). See this link for some guidelines, but typically you get a good starting position either by experience or asking for a bike fit.
Note that while saddle height is the "easiest" measurement you can setup, you also need to set the forwards/backwards position of the saddle, tilt, handlebar height and reach, etc. (which is why people do bike fits).
You should not be able to. Generally you want your leg fully extended and your foot parallel to the ground at the bottom of your stroke. As others have mentioned, there are allowances to that rule to increase maneuverability, which comes with having a lower saddle, but if you can touch the ground flat footed, and your pedal is 15-30 cm off the ground at the bottom of your stroke, then you can't possibly have an efficient stroke, and you're hurting your knees in the process.
One of the strategies to adjusting your saddle is that you ride for about 5 minutes, then raise your saddle about an inch. According to Sheldon Brown, if your bike feels lighter and faster, then that's a good adjustment for you. I raise it by inch or half inch increments, then when it feels like you're over reaching, lower it by only a quarter inch at a time. Give yourself time to adjust.