I imagine that measuring real positions and orbital elements of satellites orbiting other planets after they perform braking maneuver to get to their final orbit must be rather difficult. Especially with precision required for further maneuvering. What methods are used to measure position of probes orbiting other planets?
Here is a good tutorial on the navigation of deep space vehicles. The two main data types used are two-way Doppler (using an atomic clock reference at the DSN station, with the frequency locked to and sent back to Earth by the spacecraft), which gives the velocity component along the Earth-spacecraft line to better than 0.1 mm/s, and ranging (sending a pseudo-noise signal which is immediately reflected back by the spacecraft radio), which gives the distance from the DSN station to the spacecraft to better than one meter. Though this only gives you one component of position and one component of velocity, you can use many of these high-accuracy measurements over time combined with the constraints of motion in the gravity field of the Sun and planets to solve for the other components of position and velocity.
There is another data type, used less frequently, which is to use two widely separated DSN stations (i.e. at two different complexes on Earth) and a quasar of known position to determine the apparent position of the spacecraft with interferometry. This can determine the angular position of the spacecraft on the projection of the baseline between the two stations to an accuracy of a few nanoradians, letting you solve immediately for one more component. It is more expensive, and so is generally used when higher precision is required over a shorter time, e.g. on approach to Mars to accurately target a landing site.