TWebDBTableControl and row selection

I want to be able to keep the dataset in line with the table when a row is clicked and highlight the row. I guess that I can use OnClickCell to keep them in alignment, but how would I highlight the row?

You'd rather typically use a TWebDBGrid for this.
The TWeb(DB)TableControl is designed to visualize data but has no built-in interaction with it.
You can change the characteristics of cells (i.e. also a row) with the event OnGetCellClass where you can set CSS for each cell (thus CSS for a row you want to highlight).
All this is done automatically internally in TWebDBGrid though.