According to the OALD, group can actually serve as either a plural or singular noun. The category they give is "countable + singular or plural verb". So, an example of correct use is:
A group of us is/are going to the theatre this evening.
For any of your examples, both is and are can be used. I think the difference between them is as you say: use is when you are referring to the group, and use are when you are referring to the members of a given group. This page on when to use "group" as plural from Washington State University backs this up, saying:
When the group is being considered as a whole, it can be treated as a single entity: “the group was ready to go on stage.” But when the individuality of its members is being emphasized, “group” is plural: “the group were in disagreement about where to go for dinner.”
To analyze your given examples, here are some notes:
- A sentence is a group of words that is followed by a period. Use this form if you're talking about the group itself. That is, a sentence is a group of words followed by a period.
- A sentence is a group of words that are followed by a period. This implies that "This. Is. A. Sentence" is a sentence.
- The group of people constitutes a jury.
- The group of people constitute a jury. This would be more correct, since you are describing the nature of the group.
- A group of crows is called a 'murder'. This would be correct, since you're again describing an entire group.
- A group of crows are called a 'murder'.