1. Service agreement is a conclusive evidence about your revenue generation and dispute resolution. As you are involved in providing services to overseas clients, it is advisable to sign a service agreement. Service agreement like contract of employment contains terms and conditions in respect to nature of service, basis of remuneration, service termination clause, payments clause, liability clause etc.
2. In US it may be called professional contract.
3. You do not need service agreement for GST registration but it will support your revenue claim in support of invoices.
4. services provided by you is treated as export of services which is zero-rated. You can opt for GST without payment of taxes by filing LUT. You can only defer GST registration but it does not impact much as there will be no GST liability. For filing of GST returns, you can take the assistance of CA.
5. Once you have obtained the GST registration then it is mandatory to file GST returns irrespective of any transaction during a month.