A cedula is a national identity card. You are a DR citizen with it, entitle to most (I think) legal rights.
As for needing them, you will hear different answers from different people.
I've lived here for 2.5 years without any papers. I have a full time job at an international company, they used my passport to pay me, and wrote me a letter to get a bank account.
I doubt I will ever get any papers, unless I decided I needed a car, or learned the language well enough to work for a more "local" company.
Will every job hire you without residency? No.
If you don't speak Spanish, then your options are already limited, but since your English is a benefit to a place like a call center, you can be hired by those types of locations.
As for how to get them... .the legal section has this covered 110%, check it out.
