What are the requirements for obtaining an airworthiness bilateral agreement and how does a country obtain one?
The U.S. only enters into Bilateral Aviation Safety Agreement with Implementation Procedures for Airworthiness with countries that have an existing aircraft manufacturing industry with products to be exported to the U.S.

The process for requesting a Bilateral Aviation Safety Agreement (PDF) with Implementation Procedures for Airworthiness begins with the Foreign Ministry of the other country sending a diplomatic note to the U.S. Department of State and the FAA determining the need for an agreement. From that point the FAA begins a technical assessment, the entire process usually takes 3 to 7 years.

Is there any way for a manufacturer in a country without a bilateral agreement to import their product into the United States?
Generally speaking, there is no way for a manufacturer in a country without a bilateral agreement to directly export an aeronautical product into the U.S. for operation under a standard airworthiness certificate.

Bilateral agreements policies & guidance