Learn Dutch: Mag ik de telefoon hier gebruiken?

This is the "verb bracket" at work! When you use a modal verb, the conjugated verb (mag) stays near the front, but the main action verb (gebruiken) gets pushed all the way to the very end of the sentence.

Why is mag at the very beginning of the sentence?

Because this is a yes/no question. To turn a statement (ik mag, meaning "I may") into a yes/no question, you swap the subject and the verb. This puts mag in the first position.

Is mag just the ik form of mogen?

Yes, but it is also used for jij (you) and hij/zij/het (he/she/it). Mogen (to be allowed to) is an irregular modal verb, so its singular forms are all mag and don't follow the normal spelling rules.

Could I change the order and say "Mag ik hier de telefoon gebruiken?"

Yes, you can! While a specific object (de telefoon) often comes before a location (hier), placing hier earlier is also grammatically correct and just puts a slight emphasis on "here". The most important rule is that gebruiken must stay at the very end!