Questions & Answers about ما آتش را با شیلنگ مهار کردیم و آنها با خاک.
The second clause drops the verb entirely, but wouldn't the missing verb need a different ending for the new subject "they"?
Exactly! The implied verb is مهار کردند (they controlled). In Farsi coordinate clauses, you can safely omit the second verb to avoid repetition (a process called takhfif), even if the subject changes and the omitted verb would technically have a different personal ending than the one stated in the first clause (مهار کردیم).
The direct object آتش را (the fire) is also missing from the second half. Is that normal?
Yes, it is very common. Along with the verb, Farsi readily drops the direct object in the second clause if it is identical to the one in the first clause. The complete thought is آنها آتش را با خاک مهار کردند, but reducing it to just آنها با خاک is much more concise and sounds perfectly natural to native speakers.
Could we have used کنترل کردیم instead of مهار کردیم for "controlled"?
You would be understood, but مهار کردن is the much better choice here. It carries the specific meaning of subduing, restraining, or reigning in something wild or dangerous. It is the standard Farsi verb for dealing with fires, raging floods, wild animals, or even intense anger, whereas کنترل کردن is used for more general, everyday managing or checking.
Why does the sentence just say با شیلنگ (with hose) instead of using a word for "a" or "the" hose?
In Farsi, when you refer to using a tool in a general sense—focusing on the method rather than a specific physical object—you usually leave the noun bare. If you wanted to single out one particular hose, you could say (with a hose) or (with this hose), but the bare noun is standard for expressing the means of an action.