Learn Spanish (Latin America): Una minoría de los lomos están rotos.

This is known as agreement by sense. When a singular collective phrase like una minoría de is followed by a plural noun like los lomos, the verb usually takes the plural form (están) to match the plural noun, which is what the sentence is logically about.

Is it possible to use the singular verb está instead of están here?

Yes, strictly speaking, you could use está to match the singular noun minoría. However, when the plural noun (los lomos) is right next to the verb, using the plural verb (están) sounds much more natural and is the preferred choice in everyday Spanish.

Why is the adjective rotos masculine and plural instead of feminine and singular (rota) to match minoría?

The adjective describes what is broken. Since the spines (los lomos) are the actual things that are broken, rotos must agree with that masculine plural noun, rather than the feminine singular collective word minoría.

I usually see lomo used for pork loin or animal backs. What does it mean in this context?