Here are some bounds that I can extract from the dynamic survey [Small Ramsey Numbers][1] by Stanisław Radziszowski. 

**Claim.** 
$13 \le \mathrm{RK}_{xx} \le 18.$

*Proof.* The lower bound $13 \le \mathrm{RK}_{xx}$ was proven by Will Brian in the comments to the question.  For the upperbound, we have 
$\mathrm{RK}_{xx} \le R(K_{3,3}, K_{3,3}) = 18$.  Note that $R(K_{3,3}, K_{3,3})=18$ was proven by H. Harborth and I. Mengersen in *The Ramsey Number of $K_{3,3}$* (see Section 3.3.1 of the survey). $\Box$

**Claim.** $\mathrm{RK}_{rg} \le 62.$ 

*Proof.* $\mathrm{RK}_{rg} \le R(K_5, K_6-e) \le 62$. The bound $R(K_5, K_6-e) \le 62$ is given in Section 3.1 of the survey. $\Box$

Here is an improved bound for $\mathrm{RK}_{rg}$, using a few more results from the survey.  

**Improved Claim.** $\mathrm{RK}_{rg} \le 47$.

*Proof.* By considering the edges incident to a fixed vertex, we get the easy inductive bound $\mathrm{RK}_{rg} \le R(K_5, K_{2,3})+R(K_4, K_{3,3})+1$.  Repeating the argument again, we obtain 
$$\mathrm{RK}_{rg} \le R(K_5, K_{2,3}) + R(K_4, K_{2,3})+R(K_3, K_{3,3})+2.$$ In Section 5.9 of the survey, we have $R(B_3, K_4)=14$ and $R(B_3, K_5)=20$, where $B_3=K_2 + \overline{K_3}$.  Since $K_{2,3} \subseteq B_3$, we have $R(K_5, K_{2,3}) \le 20$, and $R(K_4, K_{2,3}) \le 14$.  Finally, in Section 3.2 of the survey, it is noted that $R(K_3, G)$ has been computed exactly for all connected graphs up to $9$ vertices.  The value of $R(K_3, K_{3,3})$ is not given explicitly in the survey, but tracking down the references, we have $R(K_3, K_{3,3})=11$.  Substituting, we obtain $\mathrm{RK}_{rg} \le 47$, as required. $\Box$ 



  [1]: https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1