Learn Portuguese (Portugal): Eles têm de suportar o frio e chegam a desmaiar. - They have to endure the cold and even faint.

You are right that chegar usually means "to arrive". However, when you use the structure chegar a followed by an infinitive verb (like desmaiar), it describes an extreme or culminating action. It means a situation went so far that a specific event happened. In English, we usually translate this as "even [doing something]" or "going as far as to [do something]". Here, it means they went as far as fainting.

Is suportar a false friend? Can I use it to say I "support" my friends?

Yes, it is often a false friend! While suportar can mean "to support" in a physical sense (like a wall supporting a roof), it usually means "to endure", "to bear", or "to tolerate" when talking about hardship, like o frio (the cold). If you want to talk about supporting a friend emotionally or supporting a sports team, you should use the verb apoiar.

Why does the sentence use têm de instead of têm que for "have to"?

In European Portuguese, ter de is the standard, grammatically preferred way to express a strong obligation or necessity ("have to" or "must"). You will often hear native speakers say ter que in everyday informal speech, but ter de is what you will typically see in writing and is considered more correct. They mean exactly the same thing.

Do I need to make desmaiar reflexive and say chegam a desmaiar-se?