Learn Danish: Får jeg lov til at låne din bil?

Both work well, but får jeg lov til literally translates to 'do I get permission to'. It is a very natural and slightly more explicit way of asking for someone's blessing. Må jeg låne din bil? is also perfectly correct and means the same thing.

Wait, doesn't låne also mean 'to lend'? How do we know it means 'borrow' here?

Yes, låne covers both 'borrow' and 'lend' in Danish! You have to rely on context. Here, because you are asking for permission for yourself (jeg), it logically means 'borrow'. If someone else was the subject, like vil du låne mig din bil? (will you lend me your car?), it translates to 'lend'.

Why is there a til right before at låne?

The fixed expression for 'to be allowed' or 'to get permission' is at få lov til. The preposition til belongs to the noun lov. When you want to specify what you are getting permission to do, you add at plus the infinitive verb, making the whole chunk får lov til at låne (get permission to borrow).

How can we tell the gender of bil from this sentence?

You can tell it is a common gender word (en bil) because it uses the possessive pronoun din (your). If the noun were neuter (an et-word), the sentence would have to use dit instead.