1) I would like to ask about the dates of the program: when does it start and what is its duration?
Exact dates will be determined by the successful applicant and the supervisor. As indicated in the website it will be 6-8 weeks, within June/July (ex. 6 weeks could be mid-June-July, or 8 weeks could be both months completely). If you should have any constraints with your schedule, this should be mentioned in your application letter.
2) With regards to the letters of reference for the Hoffleit Fellowship, should they be sent separately by the provider or altogether along with my application?
Reference letters should be uploaded by the reference writer at http://astronomy.yale.edu/hoffleitreference.
3) Should we provide transcripts for the first semester or the whole year?
You should send what is available by the time of the application deadline.
4) I would also like to know what responsibilities the post involves on my behalf and the fields I will have to work on.
Specific responsibilities will depend on the project and supervisor, but you should expect to work with real astronomical data in the area of expertise of the supervisor. Take a look at the department website and if particular areas interest you, email the project leader about your interest.
5) How do I know if I am admitted to this summer program?
Hoffleit scholars will be notified by phone or email in late March.
6) Am I eligible to apply if I am not a US citizen?
Absolutely! We welcome applicants of any citizenship.
7) Am I eligible to apply if I am in the final year of my undergraduate degree and planning to start my PhD next Fall?
You are ineligible. Only students who are in their 2nd or 3rd year of undergraduate study are eligible.