1. Has the module been testes on PrestShop 1.4.3 and is it fully functional on that new version?
Yes, it has been tested un all final relase version.
2. Is there a way of having a renewal message sent to each member at a specific time before his or her membership ends? For example, is there an executiable that can be called every 24 hours via a cron task?
The module has membership expiration reminder function that send emial to members at specificed days.
3. Just to confirm: The membership fee is handled through the cart system
Yes, it is. You need to create products at back office for each memberships and the price of the product will be the membership price.
hope this helps