My understanding is that you can not calculate homography from a single conic (conic is projection of a circle, in this case edge of the round screw). When camera calibration and homography between sensor and another plane (in you case the plane of the head of the screw) are known, you can calculate the orientation and location of the camera compared to the plane (or vice versa).
But, you can estimate homography from four known points. If you have slot/flat head screw, accurate estimation of the screw position and orientation can be very hard as the measurement error for the the corners of the slot will be relatively big compared to distance of the points. If you have cross type head, the four known points can be more spread apart and the homography estimation and therefore location/orientation estimation will be more accurate.
You can most likely combine the knowledge about the points of the slot corners and the edge of the screw (conic) to form even better estimation of the homography.
There is quite recent paper on the topic of estimating homography with different techniques. Note that there are many cases in which particular method of homography estimation does not work. For example the paper describes a method for computing homography from a point and a line, but it does not warn you that the point can not lie on the same line.
I have always recommended master thesis of Liljequist as good introduction paper about how to estimate camera location when camera calibration and homography are known. As Libor suggested, Multiple View Geometry by Hartley and Zisserman is good book about the camera geometry and algorithms related to it, but is also quite heavy compared to what you need for basic algorithms.