Is there any sin in going to the mosque closest to the home of the girl one intends to marry, instead of another mosque that is equidistant?

It is preferable for a person to pray in the mosque with the larger congregation, as a man's prayer with another man is purer than his prayer alone, and his prayer with two men is purer than his prayer with one man. The larger the number, the more beloved it is to Allah.

The fiancée is not a wife, and it is not permissible to see her, speak with her except out of necessity, or be alone with her. What the Shariah has permitted is to look at her once or twice to determine whether she is suitable as a wife or not. There is no harm in speaking with her for the same purpose or for other purposes, provided there is no seclusion.