When I trade with a villager they do not open up any new options when you complete a trade with them.

Useing version 5.0.6.

I am using version 5.1.1.0 and I also am unable to get more trades after they "lock out" their first trade. Do you have to gift/talk/etc with the villagers up to a particular heart level for them to unlock a new trade?

You need to complete the latest trade more (usually 1-5 times) before you can unlock next trade

Unable to reproduce this.

Yup. I don't have this bug.