An entry visa in your passport is relevant ONLY to your eligibility to be admitted to the U.S., not for remaining in the U.S. What matters while you are in the United States is maintaining your F-1 status for the duration of your academic program as reflected on your I-20. If your entry visa expires while you are in the U.S. you will have to obtain a valid one before re-entering the U.S. from a trip abroad. You cannot renew a visa in the U.S. but will have to apply for a new one at a U.S. consulate abroad, most likely in your home country.
Networking is the process of making connections and building relationships that provide you with information, advice, and further contacts, all of which will enhance your ability to make informed career decisions and tap into unadvertised internship/job vacancies. Networking can take place in a group or one-on-one setting.
Yes. Fellowships fund a variety of experiences for study, research, and work in the U.S. and overseas. All fellowship programs have a target audience that all students searching for fellowships have to bear in mind. International students are no different. While many fellowships are targeted at U.S. citizens only, many are not. In fact, most fellowships offered by Columbia invite applicants from any nationality.
An entry VISA is a travel document issued by a U.S. Consulate and attached to a page in your passport that is evidence of your eligibility to be admitted to the United States in a particular status with supporting documents. After admission to a program of study, International students who require an F-1 student visa to study in the U.S. apply for a certificate of eligibility called an I-20 from the International Students and Scholars office (ISSO). The I-20 is needed to apply for the entry visa to present upon arrival.
Remember to have your I-20 on your person and not packed away in checked luggage since you go through inspection before claiming bags. That said, if you forget your I-20, you should ask the immigration official for an I-515 upon entry, which will allow your admission in F-1 status for 30 days. With an F-1 visa you are normally admitted to the U.S. For 'duration of status' (D/S,) but if you are missing the I-20 it is most likely you will be admitted for 30 days with an I-515.