"Everyone is loved by someone or other"
I'm not sure how to translate this into predicate logic. At first, I said $\forall x\exists y,Lyx$ where $Lxy$ is x loves y. However, it is not true that there is someone who loves everyone. The sentence says that "everyone" is loved by someone or other (i.e. everyone in the domain is loved). Would it be correct to say $\exists x \exists y Lxy$ , but I don't think that 'x' in this case covers everyone in the domain.
You seem to be unaware of the fact that the order of the arguments changes the meaning of the statement. For example, if $a$ denotes Alice, and $b$ denotes Bob, then $Lab$ means 'Alice loves Bob', but $Lba$ means 'Bob loves Alice'.
Similarly, $\forall x \exists y Lxy$ means that everyone loves someone, but when we change the arguments, we get $\forall x \exists y Lyx$, which means that everyone is loved by someone, which is a different statement.