B-cell antibody production
Statement 1 is true and Statement 2, false.

B cells, in the absence of antigenic stimulation express surface receptors(B-cell receptors or BCRs), which look like normal secreted antibodies (but these are membrane bound and not secreted). What is amazing is that each B cell at a particular time expresses the same B cell receptor - same as in, all receptors produced off one B cell are of the same type, say, antigen A. Now, another B cell produces another type of receptors, say, those which detect Antigen B. There is not just one BCR on the B-cell, but many - just like any other cellular receptor.

Now, there are many B cells in the body. Very much like antigen A and antigen B, there so many more antigens that our body can produce antibodies against. (The source I quote below says $\ 10^{12} $) It is a wrong notion that just one little B cell is doing that all. There are specific B cells for each specific antibody.

Each B cell isn't necessarily producing a unique antibody, but yeah, specific. It isn't unique since there must be a few more similar B cells producing the same receptor against antigen A. So it is not just one single B cell producing antibody against 'A'.

How does the body produce so many types of specific proteins?

Being proteins, they must be encoded by genes. B-cells while maturing, undergo a recombination (shuffling) of their immunoglobulin genes which leads to this differential expression of receptors, hence the diversity.

For further reading on genetics of immunoglobulins and the generation of diversity, I'd suggest - http://www.ncbi.nlm.nih.gov/books/NBK26860/