Depends where you live. In large and medium cities, public defenders work out of the prosecutors office, usually across the hall in a different office. Most are paid a salary, they are not paid by the case. In small towns and rural areas, the public defender pool is generally members of the bar association and is made up of all the local lawyers in a specific area. Depending on the case load, a lawyer may be called on a few time a year to act as a PD without charge to the defendant or the public. If you feel your PD is giving you inadequate counsel, you have the right to ask for a different PD.
Yes, which usually takes you from the "pan and into the fire."
Attorneys for Indigents, like yourself, are usually paid out of a fund that is collected by the State Bar Association of your State. It is a small, scalable, fee depending upon the severity of the crime being defended. But, make no mistake, the amounts are very small, and I wouldn't even consider calling them "fees".
But, when you're poor, this is what you get. If you had money, then you could pay for a better defense. So, the old saying is true: "You get what you pay for", and in this case, you're not paying.