When you make a compound noun, the final noun is the one that determines the need for an article, regardless of the makeup and length of other elements of the compound noun.
For example, if the final noun is uncountable, no indefinite article is required: if is countable, an indefinite article is required.
Barbados sugar is dark brown
A Barbados amateur cricket team member was injured.
Regarding your sentences:
- [The] Formula One staff respond to fans' questions.
If you are talking about formula one staff in general, no indefinite article is required because in British English staff counts as plural (in Indian English it can also be singular- the equivalent of "staff member" in British English). A definite article is required if you are talking about a particular group of formula one staff (for example, those present at this event).
- The Calvin Klein team is being criticised for the new ad.
There is only one Calvin Klein team, it is specific, so a definite article is required.
- [The] Amazon.com personnel work hard on improving the service.
Same as (1).
- The Stop AIDS Project team seeks for volunteers.
There is only one Stop AIDS Project team: it is specific, so a definite article is required.