How are the speed, distance and time of India's MOM and USA's Maven Mars missions calculated?

First of all, trajectories of interplanetary missions are not designed to minimize the TRAVEL TIME but to minimize the COST, which is directly related to the fuel required to transfer the probe between the Earth and another planet. One can easily find a solution on how fast and in what direction should the probe travel, and how much time and energy will be spent after departure from the near Earth orbit by solving the gravitational force equations. But that's far not enough to launch a real probe. Scientists have built accurate kinetic models and databases of the motions of solar system objects, such as DE405, by NASA Jet Propulsion Laboratory. Some of them even take account of the perturbations of all major planets in the solar system. Engineers should also build models for the probe itself. For example, how much acceleration will the spacecraft get when turn on the booster, and how does the solar radiation pressure change the attitude angle during the trip. What's more, every probe carries at least one gyro system. Optical guiding telescopes are also widely used to calculate the attitude angle. The speed of the probe can be calculated by the Doppler shift of the radio frequency.

For more details, see the book Interplanetary Mission Design Handbook