It was a different situation as it was caused by an SPD ridden pregnancy that caused the weight gain and complete loss of fitness, but I did start with less than 10 minutes of exercise, walking to the postbox and back then lying down to recover. That quickly built up and within a couple of months of gently going a little bit further each time, I could walk a couple of miles which then meant I was getting fit enough for more mainstream exercise.
10 minutes isn't going to cure anything in itself, but it's starting point to greater benefits, and there is positive effects from 10 minutes being active compared to being sedentary. So yes, 10 minutes is worth it 
With diet, especially if life is overwhelming, make a positive change at a time. It might be something small like drinking more water.
Tiredness can be a vicious cycle. Avoiding activity and having quick fix simple carbs make it worse. Introducing more variety of protein, healthy fats, complex carbs and veg will long term help towards more stable energy levels. That doesn't have to mean lots of cooking and faddy diets. Something like porridge with frozen fruit takes the same time to make as a round of toast, but satisfies for far longer.
Good luck making positive changes in your life OP.