Learn German: Statt der Zange nimmt er das Klebeband. - Instead of the pliers, he takes the tape.

Because the preposition statt (instead of) requires the Genitive case. Since Zange is a feminine noun (die Zange), its definite article changes from die to der in the Genitive.

Could I use anstatt in this sentence instead?

Yes, absolutely. Statt and anstatt are completely interchangeable and both take the Genitive case. You can say Anstatt der Zange nimmt er das Klebeband with no change in meaning. Statt is just a bit shorter and more common in everyday speech.

Why is the verb nimmt placed before the subject er?

This is due to the German 'verb-second' rule. The entire prepositional phrase Statt der Zange takes up the first position in the sentence. To keep the conjugated verb (nimmt) in the second position, the subject (er) has to slide into the third position right after it.

Why is Zange singular here when 'pliers' is plural in English?

In English, tools with two parts like 'pliers', 'scissors', or 'tongs' are treated as plural ('a pair of pliers'). In German, die Zange is simply a singular noun. If he were leaving behind multiple pairs of pliers, you would use the plural Genitive: Statt der Zangen.