Please, please check my work. Given alphabet $\{a,b\}$. Instead asking conceptional questions to confirm my understand, I use some example.
Regular expression : $(a \cup b^*)(b \cup a^*)$
We have then empty string as an element of the language described by the regular expression since $b^*$ generates empty string. Then when it union with $a$, empty string as in the value of $(a \cup b^*)$, (am I correct?)
Q1. What should we call $(a \cup b^*)$, a set? or a regular expression? I am sure when two set union together we have a set, but $a$ is an element of the alphabet (I think). Can we just union element with a set this way?
Q2. String $a$ is also an element of the language, since the value of $(a \cup b^*)$ could be an "$a$", then $(b \cup a^*)$ could result in empty string, finally $(a \cup b^*)(b \cup a^*)$ could also result in an "$a$".
Q3. If my previous reasonings are correct, then $bbb$ is also an element of the regular expression.
You ask legitimate questions that call for clarification. The point is that it is common practice to identify mathematical objects that are formally different.
In your case, it is common practice to identify a regular expression with the regular language it represents. Thus $(a \cup b^*)(b \cup a^*)$ denotes at the same time a regular expression and a language.
The second identification (also of common practice) is to identify a word $w$ with the language $\{w\}$. The third identification (also of common practice) is to identify a letter $a$ with the word of length one $a$.
Let me now answer your questions:
Q1. I would call $(a \cup b^*)$ a language (that is, a subset of $\{a,b\}^*$).
Q2. You are right, the word $a$ belongs to the language $(a \cup b^*)(b \cup a^*)$.
Q3. You are also right, the word $bbb$ belongs to the language $(a \cup b^*)(b \cup a^*)$.