Where does Elizabeth get her powers from

Hover over to see the solution (spoiler-alert):

Booker sold her to Comstock when she was a baby. Just before the portal closed, baby-Elizabeth stuck her pinky out of the portal reaching for her father (Booker) and it got cut off as it closed.

To the best of my understanding, from playing the game, there are no stealth elements to the game except for one part.

The enemies usually don't wait in one place guarding or patrolling and "waiting for you", but are attacking you in waves, and everyone knows where you are, even if you are hidden.

A great example, is that there are levels where you can "roam" free at the start, until you steal, fire a weapon or use a vigor (even without hurting any one) and immediately all the cops in the level will come to your location and attack you, even if you were in some room alone, they will attack you.

The only exception is near the end of the game (minor spoiler ahead):

where there is an enemy with a "search light", if you are in the light for a few seconds it will attack, if you are caught for less the the time needed, it'll continue as if it didn't see you. You can run, jump, fire and do whatever you want behind its back as long as you don't get caught in its light.

From my experience, it has no stealth elements at all.

There is one thing which is the detection range of mechanized enemies, turrets, Patriots and Zepplins. They have a detection range and even a detection delay. And there is gear that affects it (probably reduces the detection range). I wouldn't call it a stealth element, because mechanical enemies are seldom alone, and once a patriot or a turret sees you, it'll know where you are and fallow/fire suppression fire, even if you are hidden from sight. This can be counted as a stealth element, but it's a very small one.