There are lots of ways to apply enough heat to polymer clay to polymerize (harden, cure) it, but baking in an oven is the most common (convection oven or convection setting on a microwave/etc is the absolute best).
Since the time necessary for complete curing depends on the thickness of the piece, the thinner the pieces are the less time they'll take (and the quicker they'll darken or burn).
However, IF the temperature actually reaching the clay can
be kept at a good temperature (not that easy to do without taking other steps though), most polymer clays can be left in that heat for hours and just get stronger and stronger from more and more polymerization.
The exceptions to that are 3 of the main Sculpey clays since they will darken more easily than other lines and brands of polymer clay (Sculpey III and SuperSculpey, with original Sculpey being even worse). Polyform suggests a higher baking temperature even for those lines because they don't think clayers will want to wait longer, but actually using a lower temp then baking for a longer time works better for them (time and temperature are interdependent for polymerization of polymer clay according to a formula you can find in the Baking
Small or thin items won't take as long as "regular size" items to cure all the way through (and btw the upper
limit for polymer clay thickness is about 1 1/4" without using an armature underneath).
Any items may also be heated higher than the setting on the oven's dial because they might be sitting on a baking surface that holds heat too much and gets hotter than the air temperature, or because they're too close to top or bottom coils, or because they're in a hot spot of the oven, or because they have parts that stick up or out into those kinds of areas.
So especially for very small or thin items, it's good to keep them away from those factors. In addition to using a shorter baking time, the items can be elevated in various ways away from baking surfaces or other situations, or be insulated (for even bad ovens), or use other techniques.
There's info on all kinds of ways to polymerize polymer clay on the Baking page of my site, including boiling certain kinds of items (and why that's controversial to some people), various kinds of baking surfaces and their advantages/disadvantages, how to insulate to avoid darkening/burning, how to test for hot spots, using an oven thermometer, the formula I mentioned above re relationship of time to temperature, etc., etc:http://glassattic.com/polymer/baking.htm
Re "hardness" after curing polymer clay you should know that since polymer clays are plastics, any thin parts of baked clay will be fairly flexible, but if thicker parts/items will be stiff. Baked polymer clay won't ever be as "hard" as ceramic/earthen clays though, but the Sculpeys do tend to be a bit harder and stiffer than the other lines of polymer clay...they sacrifice strength for that though, unless they're thick and rounded.
You can read more about all that and how to deal with it, etc, on this page if you're interested:http://glassattic.com/polymer/Characteristics.htm
(click on Strength--Rigidity,Flexibility
As for painting on top of cured polymer clay, you can do that if you want. Generally acrylic paints (2 coats) are used for regular painting though other materials/colorants can be used as long as no paints (or clear finishes) that contain petroleum-based solvents are used--those will often l immediately or eventually begin dissolving polymer clay.
Check out these two pages for paints that are used, and won't be good, for polymer clay:http://glassattic.com/polymer/paints.htm
(for painting on top, be sure to read Preparing The Clay
(also, check out Alcohol Inks
There are many other ways to give color to polymer clay though both inside the body of the clay and on the surface.
For coloring the body of the clay, check out this page:http://glassattic.com/polymer/color.htm
Some kinds of metallic or colorant effects on polymer clay can also be created with various kinds of powders (mica powders, real-metal powders, ground pigments and spices, etc), oil pastels and colored pencils, leaf or foil, and metallic waxes, to mention most of them. They can also be colored by mixing more particulate matter into the clay which are called "inclusions" (or applying them after baking mixed into a clear medium).
There's more info on all those things on these pages too (just use the alphabetical navigation bar on the Table of Contents
page, or any of the pages above, to go to them):http://glassattic.com/polymer/contents.htm
...Powders, Waxes (see the sub-category As Paint
if you're interested in that too)
...Leaf & Foil