Does the Follow Up Email extension allow sending emails to customers asking them to review products they have previously purchased, including a link to each product review form in the email template?
Yes, the extension allows doing that.
A pre-defined email template for this exact purpose comes with the extension by default, "AW Review Request" (Backend » Newsletter » Newsletter Templates)
NOTE: the Follow Up Email extension uses "double template" constructions. The "AW Review Request" template represents the general text and appearance of an outgoing email. The items listed in this template (including product details, store links, etc) are rendered by its "child" template, "AW Review Request Per Product Row".
Thus, if there's a need to change how each of the products is displayed in emails, you would need to edit the latter template.