Yes. Apparently Tom Womack used to maintain a webpage devoted to Mordell curves of high rank at the (no longer functional) page here although it can be viewed with the help of the Wayback Machine. This reports that the curves with $$ k = 35470887868736225 \text{ and } k = -46111487743732324 $$ have rank $11$, while the curve with $$ k = -6533891544658786928 $$ has rank $12$. This rank 12 curve was found by Jordi Quer in connection with searching for imaginary quadratic fields of high $3$-rank (see the paper ''Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12'' in C. R. Acad. Sc. Paris. from 1987).

Elkies and Rodgers also found a rank $11$ curve of the form $x^{3} + y^{3} = k$ (isomorphic to $y^{2} = x^{3} - 432k^{2}$) in their 2004 ANTS paper.

I do not know if any of these results have been improved recently.

Yes. Apparently Tom Womack used to maintain a webpage devoted to Mordell curves of high rank at the (no longer functional) page here although it can be viewed with the help of the Wayback Machine. This reports that the curves with $$ k = 35470887868736225 \text{ and } k = -46111487743732324 $$ have rank $11$, while the curve with $$ k = -6533891544658786928 $$ has rank $12$. This rank 12 curve was found by Jordi Quer in connection with searching for imaginary quadratic fields of high $3$-rank (see the paper ''Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12'' in C. R. Acad. Sc. Paris. from 1987). Quer reports two more rank 12 Mordell curves in this paper.

Elkies and Rodgers also found a rank $11$ curve of the form $x^{3} + y^{3} = k$ (isomorphic to $y^{2} = x^{3} - 432k^{2}$) in their 2004 ANTS paper.

I do not know if any of these results have been improved recently.