Learn Turkish: Balon artık giderek alçalıyor. - The balloon is now gradually descending.

Giderek is made of the verb gitmek (to go) and the adverbial suffix -(y)erek (by doing / while doing). Its literal meaning is "by going" or "as it goes." However, it is used as a fixed expression in Turkish to mean "gradually" or "more and more."

Could we use gittikçe instead of giderek here?

Yes! You can absolutely say Balon artık gittikçe alçalıyor. Both gittikçe and giderek are interchangeable when expressing a gradual change or an ongoing process.

Why use artık here instead of şimdi? Both can mean "now".

Şimdi refers simply to the present moment ("right now"). Artık implies a change in state or situation, meaning "from now on" or "at this point." Using artık emphasizes that the balloon wasn't descending earlier, but now it has started to.

What is the root of the verb alçalıyor?

It comes from the adjective alçak (low). When the verb-forming suffix -l is added to create alçalmak (to become low / to descend), the final -k is dropped. You will see this exact same pattern with yüksek (high) becoming yükselmek (to rise).