Learn Farsi: مبارزه کردن یک هنر قدیمی است.

In Farsi, you can use the full infinitive form of a verb to act as a noun, just like adding the -ing ending to a verb in English to make a gerund. Here, the long infinitive mobaarezeh kardan acts as the subject of the sentence.

Since mobaarezeh is already a noun meaning struggle or combat, why do we need kardan here?

While mobaarezeh is a noun, adding the infinitive kardan focuses on the active process or action of doing it. Together, the compound infinitive mobaarezeh kardan perfectly translates to the active concept of fighting as a verbal noun.

Where is the hidden ezafe in this sentence?

There is an ezafe linking the noun honar to its adjective qadimi. When spoken, it sounds like honar-e qadimi. Remember that in Farsi, adjectives always come after the noun they describe.

Why doesn't the word yek get connected with an ezafe too?

Numbers and quantifiers like yek come right before the noun and do not take an ezafe. You simply say yek honar, and only use the ezafe to attach the descriptive adjective that comes after the noun.