Transitive definition is $\forall x[(x,y)∈R \land (y,z)∈R \implies (x,z)∈ R]$.
If $(a,b), (b,a) ∈ R$, then can I say that $(a,a) ∈ R$?
Because based on the definition $x,y,z$ are not the same value. So, I am not sure whether $(a,a)$ can be a transitive.
The definition of transitive for relations applies to all $x,y,z$ in the domain of the relation. It does not require them to be distinct. So if $R$ is a transitive relation, then the reasoning $(a,b),(b,a) \in R$ implies $(a,a) \in R$ is valid.
Note however that a relation $R$ being transitive and symmetric does not necessarily imply that $R$ is reflexive, the sort of thing your argument seems intended to show. The gap in such a proof is that for some $a$ in the domain of $R$, there may not be any pair $(a,b)\in R$ and so the argument fails to get the premise it needs to start with.