Learn German: Er hat sechzehn Fehler gemacht.

You're right to notice that! When combining sechs (six) and zehn (ten) to make sixteen, German drops the 's' to make it easier to pronounce. A similar thing happens with seventeen, where sieben becomes siebzehn (dropping the 'en').

The translation says "mistakes", but isn't Fehler singular?

It is plural here! In German, many masculine and neuter nouns that end in -er (like der Fehler) do not change their spelling in the plural. You can tell it means "mistakes" instead of "mistake" because of the number sechzehn right in front of it.

Why are the verbs hat and gemacht separated by the other words?

This is a classic German sentence structure for the past tense. When using the spoken past tense, the helping verb (hat) takes the normal second position in the sentence, but the main past action (gemacht) is pushed all the way to the very end.

How do we get gemacht from the verb machen?

Gemacht is the past participle of machen (to make or do). For regular verbs in German, you build this past form by adding ge- to the beginning and -t to the end of the verb stem (mach).