1. What does it mean when the confidence interval for the difference between two means includes zero?
If the interval includes 0, then this means that the difference between the means of the populations could be 0 and therefore the means could be the same.
If the interval does not include zero, then it is unlikely that the means of the populations will be the same.
2. What degree of confidence should I use when constructing a confidence interval?
Usually in a question for an examination the required degree of confidence is given.
In situations involving important issues such as medical matters or large amounts of money it is better to be cautious and use high levels of confidence e.g. 99%. This provides a wider interval.