Does not using the brain decreases mental functioning? If so, is it recommended to keep using the brain, for example by learning new things or engaging in challenging activities (puzzels, quizzes etc.) up until old age?
Yes, brain power is eroded via a lack of practice. This occurs through the processes called synaptic pruning and brain plasticity.
I will leave you with a very basic answer as I am unsure of your level of understanding of cognitive processes. First, you must understand neurons. Then, you can begin to understand synaptic pruning and brain plasticity. Finally, here is a link to a bunch more information and brain plasticity exercises.
Not using your brain might well be deleterious, but it's impossible not to use your brain unless you're in a coma or something. There are some cool studies on plasticity (see Shayna's answer above) in amputees, where the parts of the brain that control the amputated limb go unused but are taken up by other functions instead.
However, "perpetual brain exercises" like the brain training games/puzzles you see have shown no efficacy in actually increasing cognitive performance on any task, besides the specific game/puzzle you're playing. Playing a lot of chess does not make your brain any more powerful, it just makes you better at chess. See the following: