>>Is the universe really billions of years old?
It's thought to be around 13.8 billion years old, yes.
>>If it is, how is Mars losing it's atmosphere?
The two points don't truly relate, however, Mars' core solidified about 3 to 4 billion years ago, and the magnetic field Mars was thought to possess died away; this allows solar wind to strip the atmosphere.
>>Mars does not have a magnetic field and cannot defend itself against
>>solar winds which bombard the planet.
That's true - which is why the atmosphere is being stripped. What was thought to be a relatively thick blanket of air is now less than 1% as thick of Earth's.
>>Nasa's MAVEN probe found that Mars loses 100 grams of it's atmosphere per second.
At current rates; when it was thicker, and core had died, it probably lost more, quicker.
>>In addition, during solar flares the rate increases by a factor of 10.
Interesting - but, that's when a flare's plume sweeps over the planet; it's not every single solar flare...
>>Mars must be a young planet!
It's younger than the universe to be sure, but it's not exactly known when it's atmosphere began to be stripped. You can't deduce the age of the planet by the thinness of it's atmosphere.