5 and 7 point scales are standard, though use of more categories might be justified if responses can be expected to lie mostly at one end. For example, the statement “the most appropriate role for women in society is in the home” could be expected to generate mostly “disagree” responses in western societies, and the use of a larger number of categories would help distinguish shades of difference.
Whether to include a neutral midpoint has been a subject of disagreement for years. What does seem clear is that inclusion of a neutral category can shift the overall negative-positive balance. Researchers have done A-B tests in which a sample is split randomly, with each half receiving the same statement wordings except that one of the groups has an added neutral category on their Likert scale. The problem is that some of these A-B tests have shown the neutral category inclusion shifts opinion in a positive direction, while other A-B tests have shown the reverse. So it seems to depend on the situation and the question being asked.