«Есть» is not used here because existence/having is not the main point this sentence makes. The fact that mom has a voice is implied, it's not the point of the sentence. The point of the sentence is that mother's voice is loud.
Yes, «у ма́мы есть го́лос» is correct. Without a context, «у мамы есть голос» is likely to be understood that mum has a good vocal abilities, that she can sing well.
No. Russian expresses possession differently from English: we don't usually use the verb 'to have'. Instead, we say that something exist and use «у» preposition (with genitive) to indicate possessor. So, «у ма́мы гро́мкий го́лос» liteally means something like 'at mother['s possession, there] is [a] loud voice'.
When you make a comparison, you follow the same grammar. So, «У мамы́ гро́мкий го́лос, как у быка́» means 'at mother's [possession, there] is [a] loud voice, like at bull's [possession]'.
If you use nominative, the sentence gets a totally different meaning: «у ма́мы гро́мкий го́лос, как бык» means that mother's voice is similar to a bull: "at mother's [possession, there] is [a] loud voice, like [a] bull'. But how can voice be similar to a bull? 'Loud voice, like a bull' sounds very strange.
i think the article doesn't go for countable/uncountable nouns. instead, it goes to definite/indefinite one. example for "a man" vs "the man". "a man" could be anybody, while "the man" refers to someone that we already knew. i am not a native english speaker so feel free to correct me.