theres generally two ways that physics engines are built:

one is that time is progressed in a binary search style to each exact time of collisions and impulses are applied.
the problem with this is such as you have described, but mainly what happens when impulses become very small, and numerical innacuracies means the object is constantly colliding and slowing down simulation eventually making it stuck? You get to a point of consistent contacts which have to be treat seperately and for realism involves solving a large number of equations simultaneously, dynamically. I havn't had much experiance in this method for that main reason, it seems too messy.

The other way, the way i tend to use is a penetration method. you time-step your objects however you may be doing it, and you go through each collision and find the minimal distance (along a penetration axis -> in 2D the unique normals of the shapes (for polygons) ) and move the objects apart by this value (dependant on masses for the distribution) and then idealized impulses (i.e. step change impulse)'s are applied with various models of friction, the disadvantage is that for fast moving objects it is not PERFECT, and also that with stacked objects, you require multiple collision passes (ive only ever needed between 2 and 10) for no penetrations to persist.

the problem you seem to be describing is a result of the first method as hinted by your second paragraph.

if your consistent in using this method, myphysicslab has an article on handling resting contacts which is a little above me at the moment, but you may be able to work through it: http://www.myphysicslab.com/contact.html

yes, you use seperating axis theorem for the penetration method.

the most ive ever experimented with no polygonal objects is with circles wherein you can probably imagine what axi you would use, but otherwise you really kind of have to use a good approximation , i mean even proper commercial games will use approximations for smooth surfaces.