How would the addition of a second sun, located at the other focus of the planet's elliptical orbit, impact the motion and orbit of the planet? Specifically, how would the gravitational forces from both suns combine to influence the planet's trajectory?
It will depend a lot on the mass of the second sun and the eccentricity of the orbit of the planet. Quick and dirty approximate solution would be that for most of the planets in our solar system (in low eccentricity orbits) it won't be all that much different to adding an extra solar mass to the original central sun and shifting it suddenly to the mid point of the two focii.
The main first order effect would be that the planets would all find themselves in much more eccentric orbits since their orbital speed is no longer enough to maintain a nearly circular orbit at their previous distance from the sun. (we'll ignore for now the practical difficulties of conjuring up a solar mass star from thin vacuum)
Three body systems can be remarkably complex dynamical systems and subject to a great deal of study. Wiki isn't too bad on the basics of the Three Body Problem.
A much safer situation for a planet that you were fond of is to put a second sun so as to form an equilateral triangle in the same orbital plane with the original sun-planet line and a similar orbital velocity. The Trojan asteroids share their orbit with Jupiter in just such a configuration.
A close double star can have planets orbiting it provided that they are not in the zones where resonances prevent long term stable orbits. Jupiter (and to a much lesser extent) Saturn fulfil that filtering role in our own solar system. A planet which found itself in that awkward situation after the sudden change would slowly be pumped into another orbit (or if very unlucky get ejected).
Mercury will be in big trouble though since its orbit is moderately eccentric ~0.205 and rather too close to the sun. Pluto although in an eccentric ~0.25 orbit is far enough away that things will pretty much average out.