It is much more efficient to go from concrete to general here. Instead of us giving every possible variation/example and then you saying what you need it for, you can tell us how you want to use it and then we can go from there.
BTW I don't know what discipline you're in but you don't need to have boring headings unless it's required. Usually boring headings like that are for the proposals.
It doesn't sound like English. It's not something that I have ever heard or read. But 'statement' + noun works here, statement of intent, statement of purpose.
It's not mandatory, but accepted. It saves time, space and money. Telegraphs used to be very expensive, and newsprint is also costly, so saving space is a good idea.