Learn Farsi: بعد از تزریق، نفس او آرام شد. - After the injection his breathing became calm.

Yes! If you swap them to say به محضِ تزریق (be mahz-e tazriq), it changes the meaning from a general "After the injection" to "Immediately after the injection" or "Upon injection." Both phrases work perfectly with the abstract noun تزریق (injection).

Do I need an Ezafe (an "-e" sound) after از to connect it to تزریق?

No, the preposition از (az) never takes an Ezafe. You simply say ba'd az tazriq. This is an important contrast to the lesson's target phrase به محضِ (be mahz-e), where the noun محض must take an Ezafe to link to the following word.

نفس usually means "breath." Why does the English translation say "breathing"?

In Farsi, نفس (nafas) covers both a singular "breath" and the continuous state of "breathing." While you could use the more clinical Arabic-derived verbal noun تنفس (tanaffos) for "respiration," نفس is the standard, natural choice here. Remember to pronounce the hidden Ezafe connecting it to the pronoun: nafas-e u (his breathing).

Could I use the verb گرفتن (to take) here instead of شدن (to become)?

Yes, آرام گرفت (aram gereft) is a very common and idiomatic alternative. While آرام شد simply means "became calm," the compound verb آرام گرفتن implies "settling down" or "stabilizing," which perfectly describes a physical state like breathing or a heart rate returning to normal.