Even with proper warm-up and cool-down, it is normal for your calf pain and tiredness due to lactic acid accumulation. This feeling will fade out after jogging for 1-2 weeks after your body adapting to the exercise.
Although it is more effective for weight losing when jogging for >30 (or 40) minutes, you should add your jogging time step by step. It is ok for 3-4 times/week initially but it will be better to add to 5 times/week after your adaptation.
I think there are many ching and ctse here would give you more useful advice!