Decidability of language of TMs which accept only their Gödel number
The recursion theorem states that if $A(x,y)$ is a two-input Turing machine then there is a Turing machine $M$ such that $M(x) = A(x,\langle M \rangle)$. Moreover, $M$ can be constructed from $A$ effectively (i.e., using an algorithm). Using this, we can reduce the halting problem to your problem. Given a Turing machine $B$ and an input $z$, construct a new Turing machine $A(x,y)$ which acts as follows:


If $x = y$ then simulate $B$ on $z$.
Otherwise, enter an infinite loop.


Use the recursion theorem to construct a new machine $M$ that on input $x$ acts as follows:


If $x = \langle M \rangle$ then simulate $B$ on $z$.
Otherwise, enter an infinite loop.


Then $M \in L$ iff $B$ accepts $z$.