I recommend that you become familiar with graphical notation. As you will see from my answer, other ("formula") approaches are sub-par.
Formula (4.1) on page 14 of the linked paper by Kirillov defines a morphism $T_X : A \to A$ as follows. First, $A$ is a rigid algebra object in a modular category $\mathcal C$ (with all associators suppressed, and braiding denoted $\beta_{M,N} : M\otimes N \to N\otimes M$), and $X$ is a rigid $A$-module. I will write the multiplication on $A$ by $m_A$ and the left-action of $A$ on $X$ by $m_X$. Not having read the paper carefully, I believe that "rigid" means that $A$ and $X$ are each isomorphic to their own duals, and that this isomorphism is chosen to have good compatibility properties. In particular, $A$ should in fact be a Frobenius algebra for this isomorphism, and perhaps a symmetric one at that. I will write the unit as $u_X : 1 \to X\otimes X$ and the counit as $\epsilon_X : X\otimes X \to 1$, and similarly for $A$. Then we can consider the following composition:
$$ \begin{eqnarray} A & \to & X \otimes X \otimes A & \quad\quad & (u_X \otimes \mathrm{id}_A) \\
& \to & X \otimes A \otimes X && (\mathrm{id}_X \otimes \beta_{X,A}) \\
& \to & X \otimes A \otimes A \otimes A \otimes X && (\mathrm{id}_A \otimes \mathrm{id}_X \otimes u_A \otimes \mathrm{id}_{X}) \\
& \to & X \otimes A \otimes X && (\mathrm{id}_X \otimes m_A \otimes m_X) \\
& \to & X \otimes X \otimes A && (\mathrm{id}_X \otimes \beta_{A,X}) \\
& \to & A && (\epsilon_X \otimes \mathrm{id}_A)
\end{eqnarray} $$
Or, to put it another way,
$$ T_X = (\epsilon_X \otimes \mathrm{id}_A) \circ (\mathrm{id}_X \otimes \beta_{A,X}) \circ (\mathrm{id}_X \otimes m_A \otimes m_X) \circ (\mathrm{id}_A \otimes \mathrm{id}_X \otimes u_A \otimes \mathrm{id}_{X}) \circ (\mathrm{id}_X \otimes \beta_{X,A}) $$
The point is, this is a mess, and is entirely unenlightening what's really happening. Moreover, the coherency axioms assure that there are many equivalent ways to write the above map. For example, I could have replaced the last two steps $(\epsilon_X \otimes \mathrm{id}_A) \circ (\mathrm{id}_X \otimes \beta_{A,X})$ with $(\mathrm{id}_A \otimes \epsilon_X) \circ (\beta_{X,A}^{-1} \otimes \mathrm{id}_X)$.
As for your second question, skimming did not for me reveal any place in the paper where that notation is used. I could make guesses, but perhaps someone else will have studied this paper more closely.
A final, non-math remark: The phrase "punctured curve" has a technical meaning in various areas, including areas close to this paper, to mean a compact Riemann surface with finitely many points removed (or variations on this notion). The standard term for "font" in which Kirillov draws his $A$ strands is "dashed", as opposed to "solid" for $X$. And a good term for the edges in such diagrams is "strands" — I have also seen "edges" and "strings", but the latter in particular is problematic because to a physicist a "string" is something that through time traces out a surface ("worldsheet"), whereas one meaning of these graphical calculi is some "particles" $A$ and $X$ traveling through time and thereby tracing out "worldlines".
