Normally acceptance tests and behavior tests focus on making sure value is delivered because by their own nature they are a form of black box testing.
So for 1. the answer is NO, they should not be complete. They should make sure that the external behaviors that produce value do not regress.
Regarding 2. I would avoid such tools, because they are devilishly hard to query to get time based information: typically agile tools like Rally or Version one are able to do burn down charts to give you burn downs by day and velocity charts. Use bug trackers for tracks and agile tools for agile!