Learn English – Is the use of “alternative options” redundant in this sentence

Yes, it is redundant since "currently" and "at the moment" mean the same thing. Instead, you could say either:

We're currently overbooked.

or

We're overbooked at the moment.

This comes up a lot with cousin as many other languages have more words for different types of cousin than English, though not always in the same way as you say for the Chinese languages.

Generally, we just say "cousin" unless it's particularly relevant. If it was relevant we might be happy enough that the subsequent her does indicate her being female.

We might include it if it was particularly relevant, even given the subsequent information from her. E.g. if you suspected discrimination.

But in terms of translation, normally the natural way to talk about any such relative is just a bare "cousin".